Statistics of the multiplicative group Greg Martin University of - PowerPoint PPT Presentation

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Number theory within a computation of M n Example: M n when n = 11 ! = 2 8 · 3 4 · 5 2 · 7 · 11 M 11 ! ∼ = M 2 8 × M 3 4 × M 5 2 × M 7 × M 11 ∼ = ( C 2 ⊕ C 64 ) ⊕ C 54 ⊕ C 20 ⊕ C 6 ⊕ C 10 ∼ = ( C 2 ⊕ C 64 ) ⊕ ( C 2 ⊕ C 27 ) ⊕ ( C 4 ⊕ C 5 ) ⊕ ( C 2 ⊕ C 3 ) ⊕ ( C 2 ⊕ C 5 ) ∼ = C 2 ⊕ C 2 ⊕ C 2 ⊕ C 2 ⊕ ( C 4 ⊕ C 3 ⊕ C 5 ) ⊕ ( C 64 ⊕ C 27 ⊕ C 5 ) ∼ = C 2 ⊕ C 2 ⊕ C 2 ⊕ C 2 ⊕ C 60 ⊕ C 8 , 640 Largest invariant factor 8 , 640 = λ ( 11 !) (Carmichael lambda-function) Number of invariant factors When n odd: exactly ω ( n ) = # { p | n } When n even: ω ( n ) − 1 or ω ( n ) or ω ( n ) + 1 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Number theory within a computation of M n Example: M n when n = 11 ! = 2 8 · 3 4 · 5 2 · 7 · 11 M 11 ! ∼ = M 2 8 × M 3 4 × M 5 2 × M 7 × M 11 ∼ = ( C 2 ⊕ C 64 ) ⊕ C 54 ⊕ C 20 ⊕ C 6 ⊕ C 10 ∼ = ( C 2 ⊕ C 64 ) ⊕ ( C 2 ⊕ C 27 ) ⊕ ( C 4 ⊕ C 5 ) ⊕ ( C 2 ⊕ C 3 ) ⊕ ( C 2 ⊕ C 5 ) ∼ = C 2 ⊕ C 2 ⊕ C 2 ⊕ C 2 ⊕ ( C 4 ⊕ C 3 ⊕ C 5 ) ⊕ ( C 64 ⊕ C 27 ⊕ C 5 ) ∼ = C 2 ⊕ C 2 ⊕ C 2 ⊕ C 2 ⊕ C 60 ⊕ C 8 , 640 Largest invariant factor 8 , 640 = λ ( 11 !) (Carmichael lambda-function) Number of invariant factors When n odd: exactly ω ( n ) = # { p | n } When n even: ω ( n ) − 1 or ω ( n ) or ω ( n ) + 1 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n What’s known about λ ( n ) For purposes of comparison φ ( n ) is never smaller than ( e − γ + o ( 1 )) n / log log n . Theorem (Erd˝ os/Pomerance/Schmutz, 1991) For almost all integers n , � � λ ( n ) = n / exp ( 1 + o ( 1 )) log n log log n . � � In other words, the normal order of log n /λ ( n ) is log n log log n . Drive-by question What about the second-largest invariant factor, λ 2 ( n ) ? k ≥ 1 : a k ∈ � 1 � (mod n ) for all ( a , n ) = 1 } . � λ ( n ) = min Pick a reduced residue a 1 of order λ ( n ) . Then k ≥ 1 : a k ∈ � 1 , a 1 � (mod n ) for all ( a , n ) = 1 } . � λ 2 ( n ) = min Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n What’s known about λ ( n ) For purposes of comparison φ ( n ) is never smaller than ( e − γ + o ( 1 )) n / log log n . Theorem (Erd˝ os/Pomerance/Schmutz, 1991) For almost all integers n , � � λ ( n ) = n / exp ( 1 + o ( 1 )) log n log log n . � � In other words, the normal order of log n /λ ( n ) is log n log log n . Drive-by question What about the second-largest invariant factor, λ 2 ( n ) ? k ≥ 1 : a k ∈ � 1 � (mod n ) for all ( a , n ) = 1 } . � λ ( n ) = min Pick a reduced residue a 1 of order λ ( n ) . Then k ≥ 1 : a k ∈ � 1 , a 1 � (mod n ) for all ( a , n ) = 1 } . � λ 2 ( n ) = min Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n What’s known about λ ( n ) For purposes of comparison φ ( n ) is never smaller than ( e − γ + o ( 1 )) n / log log n . Theorem (Erd˝ os/Pomerance/Schmutz, 1991) For almost all integers n , � � λ ( n ) = n / exp ( 1 + o ( 1 )) log n log log n . � � In other words, the normal order of log n /λ ( n ) is log n log log n . Drive-by question What about the second-largest invariant factor, λ 2 ( n ) ? k ≥ 1 : a k ∈ � 1 � (mod n ) for all ( a , n ) = 1 } . � λ ( n ) = min Pick a reduced residue a 1 of order λ ( n ) . Then k ≥ 1 : a k ∈ � 1 , a 1 � (mod n ) for all ( a , n ) = 1 } . � λ 2 ( n ) = min Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n What’s known about λ ( n ) For purposes of comparison φ ( n ) is never smaller than ( e − γ + o ( 1 )) n / log log n . Theorem (Erd˝ os/Pomerance/Schmutz, 1991) For almost all integers n , � � λ ( n ) = n / exp ( 1 + o ( 1 )) log n log log n . � � In other words, the normal order of log n /λ ( n ) is log n log log n . Drive-by question What about the second-largest invariant factor, λ 2 ( n ) ? k ≥ 1 : a k ∈ � 1 � (mod n ) for all ( a , n ) = 1 } . � λ ( n ) = min Pick a reduced residue a 1 of order λ ( n ) . Then k ≥ 1 : a k ∈ � 1 , a 1 � (mod n ) for all ( a , n ) = 1 } . � λ 2 ( n ) = min Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n What’s known about λ ( n ) For purposes of comparison φ ( n ) is never smaller than ( e − γ + o ( 1 )) n / log log n . Theorem (Erd˝ os/Pomerance/Schmutz, 1991) For almost all integers n , � � λ ( n ) = n / exp ( 1 + o ( 1 )) log n log log n . � � In other words, the normal order of log n /λ ( n ) is log n log log n . Drive-by question What about the second-largest invariant factor, λ 2 ( n ) ? k ≥ 1 : a k ∈ � 1 � (mod n ) for all ( a , n ) = 1 } . � λ ( n ) = min Pick a reduced residue a 1 of order λ ( n ) . Then k ≥ 1 : a k ∈ � 1 , a 1 � (mod n ) for all ( a , n ) = 1 } . � λ 2 ( n ) = min Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n What’s known about λ ( n ) For purposes of comparison φ ( n ) is never smaller than ( e − γ + o ( 1 )) n / log log n . Theorem (Erd˝ os/Pomerance/Schmutz, 1991) For almost all integers n , � � λ ( n ) = n / exp ( 1 + o ( 1 )) log n log log n . � � In other words, the normal order of log n /λ ( n ) is log n log log n . Drive-by question What about the second-largest invariant factor, λ 2 ( n ) ? k ≥ 1 : a k ∈ � 1 � (mod n ) for all ( a , n ) = 1 } . � λ ( n ) = min Pick a reduced residue a 1 of order λ ( n ) . Then k ≥ 1 : a k ∈ � 1 , a 1 � (mod n ) for all ( a , n ) = 1 } . � λ 2 ( n ) = min Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Distribution results: different strengths By way of analogy: some historical results about the distribution of ω ( n ) , the number of distinct prime factors of n . The average value of ω ( n ) is log log n . — requires an asymptotic formula for � n ≤ x ω ( n ) The normal order (typical size) of ω ( n ) is log log n . � 2 � — requires estimate for variance � ω ( n ) − log log n n ≤ x Erd˝ os–Kac theorem: ω ( n ) is asymptotically distributed like a normal random variable with mean log log n and variance log log n . (More precise statement on next slide.) — requires asymptotic formulas for all central moments � k � � ω ( n ) − log log n n ≤ x Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Distribution results: different strengths By way of analogy: some historical results about the distribution of ω ( n ) , the number of distinct prime factors of n . The average value of ω ( n ) is log log n . — requires an asymptotic formula for � n ≤ x ω ( n ) The normal order (typical size) of ω ( n ) is log log n . � 2 � — requires estimate for variance � ω ( n ) − log log n n ≤ x Erd˝ os–Kac theorem: ω ( n ) is asymptotically distributed like a normal random variable with mean log log n and variance log log n . (More precise statement on next slide.) — requires asymptotic formulas for all central moments � k � � ω ( n ) − log log n n ≤ x Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Distribution results: different strengths By way of analogy: some historical results about the distribution of ω ( n ) , the number of distinct prime factors of n . The average value of ω ( n ) is log log n . — requires an asymptotic formula for � n ≤ x ω ( n ) The normal order (typical size) of ω ( n ) is log log n . � 2 � — requires estimate for variance � ω ( n ) − log log n n ≤ x Erd˝ os–Kac theorem: ω ( n ) is asymptotically distributed like a normal random variable with mean log log n and variance log log n . (More precise statement on next slide.) — requires asymptotic formulas for all central moments � k � � ω ( n ) − log log n n ≤ x Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Distribution results: different strengths By way of analogy: some historical results about the distribution of ω ( n ) , the number of distinct prime factors of n . The average value of ω ( n ) is log log n . — requires an asymptotic formula for � n ≤ x ω ( n ) The normal order (typical size) of ω ( n ) is log log n . � 2 � — requires estimate for variance � ω ( n ) − log log n n ≤ x Erd˝ os–Kac theorem: ω ( n ) is asymptotically distributed like a normal random variable with mean log log n and variance log log n . (More precise statement on next slide.) — requires asymptotic formulas for all central moments � k � � ω ( n ) − log log n n ≤ x Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws Definition A function f ( n ) satisfies an Erd˝ os–Kac law with mean µ ( n ) and variance σ 2 ( n ) if � u � n ≤ x : f ( n ) − µ ( n ) � 1 1 e − t 2 / 2 dt x # < u = √ lim σ ( n ) x →∞ 2 π −∞ for every real number u . Standard notation ω ( n ) is the number of distinct prime factors of n . Ω( n ) is the number of prime factors of n counted with multiplicity. Theorem (Erd˝ os–Kac, 1940) Both ω ( n ) and Ω( n ) satisfy Erd˝ os–Kac laws with mean log log n and variance log log n . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws Definition A function f ( n ) satisfies an Erd˝ os–Kac law with mean µ ( n ) and variance σ 2 ( n ) if � u � n ≤ x : f ( n ) − µ ( n ) � 1 1 e − t 2 / 2 dt x # < u = √ lim σ ( n ) x →∞ 2 π −∞ for every real number u . Standard notation ω ( n ) is the number of distinct prime factors of n . Ω( n ) is the number of prime factors of n counted with multiplicity. Theorem (Erd˝ os–Kac, 1940) Both ω ( n ) and Ω( n ) satisfy Erd˝ os–Kac laws with mean log log n and variance log log n . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws Definition A function f ( n ) satisfies an Erd˝ os–Kac law with mean µ ( n ) and variance σ 2 ( n ) if � u � n ≤ x : f ( n ) − µ ( n ) � 1 1 e − t 2 / 2 dt x # < u = √ lim σ ( n ) x →∞ 2 π −∞ for every real number u . Standard notation ω ( n ) is the number of distinct prime factors of n . Ω( n ) is the number of prime factors of n counted with multiplicity. Theorem (Erd˝ os–Kac, 1940) Both ω ( n ) and Ω( n ) satisfy Erd˝ os–Kac laws with mean log log n and variance log log n . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws Definition A function f ( n ) satisfies an Erd˝ os–Kac law with mean µ ( n ) and variance σ 2 ( n ) if � u � n ≤ x : f ( n ) − µ ( n ) � 1 1 e − t 2 / 2 dt x # < u = √ lim σ ( n ) x →∞ 2 π −∞ for every real number u . Standard notation ω ( n ) is the number of distinct prime factors of n . Ω( n ) is the number of prime factors of n counted with multiplicity. Theorem (Erd˝ os–Kac, 1940) Both ω ( n ) and Ω( n ) satisfy Erd˝ os–Kac laws with mean log log n and variance log log n . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws Definition A function f ( n ) satisfies an Erd˝ os–Kac law with mean µ ( n ) and variance σ 2 ( n ) if � u � n ≤ x : f ( n ) − µ ( n ) � 1 1 e − t 2 / 2 dt x # < u = √ lim σ ( n ) x →∞ 2 π −∞ for every real number u . Standard notation ω ( n ) is the number of distinct prime factors of n . Ω( n ) is the number of prime factors of n counted with multiplicity. Theorem (Erd˝ os–Kac, 1940) Both ω ( n ) and Ω( n ) satisfy Erd˝ os–Kac laws with mean log log n and variance log log n . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws Definition A function f ( n ) satisfies an Erd˝ os–Kac law with mean µ ( n ) and variance σ 2 ( n ) if � u � n ≤ x : f ( n ) − µ ( n ) � 1 1 e − t 2 / 2 dt x # < u = √ lim σ ( n ) x →∞ 2 π −∞ for every real number u . Standard notation ω ( n ) is the number of distinct prime factors of n . Ω( n ) is the number of prime factors of n counted with multiplicity. Theorem (Erd˝ os–Kac, 1940) Both ω ( n ) and Ω( n ) satisfy Erd˝ os–Kac laws with mean log log n and variance log log n . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Other functions with Erd˝ os–Kac laws The paper of Erd˝ os–Kac establishes these normal-distribution laws for a large class of additive functions: if n = p r 1 1 · · · p r k k , then f ( n ) = f ( p r 1 1 ) + · · · + f ( p r k k ) . Examples of non-additive functions: Liu (2007) On GRH, ω (# E ( F p )) satisfies an Erd˝ os–Kac law with mean log log p and variance log log p . Erd˝ os–Pomerance (1985) ω ( φ ( n )) and Ω( φ ( n )) satisfy Erd˝ os– Kac laws with mean 2 ( log log n ) 2 and variance 1 1 3 ( log log n ) 3 . Ω( φ ( n )) is not additive, but is “ φ -additive”: if φ ( n ) = p r 1 1 · · · p r k k , then Ω( φ ( n )) = Ω( p r 1 1 ) + · · · + Ω( p r k k ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Other functions with Erd˝ os–Kac laws The paper of Erd˝ os–Kac establishes these normal-distribution laws for a large class of additive functions: if n = p r 1 1 · · · p r k k , then f ( n ) = f ( p r 1 1 ) + · · · + f ( p r k k ) . Examples of non-additive functions: Liu (2007) On GRH, ω (# E ( F p )) satisfies an Erd˝ os–Kac law with mean log log p and variance log log p . Erd˝ os–Pomerance (1985) ω ( φ ( n )) and Ω( φ ( n )) satisfy Erd˝ os– Kac laws with mean 2 ( log log n ) 2 and variance 1 1 3 ( log log n ) 3 . Ω( φ ( n )) is not additive, but is “ φ -additive”: if φ ( n ) = p r 1 1 · · · p r k k , then Ω( φ ( n )) = Ω( p r 1 1 ) + · · · + Ω( p r k k ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Other functions with Erd˝ os–Kac laws The paper of Erd˝ os–Kac establishes these normal-distribution laws for a large class of additive functions: if n = p r 1 1 · · · p r k k , then f ( n ) = f ( p r 1 1 ) + · · · + f ( p r k k ) . Examples of non-additive functions: Liu (2007) On GRH, ω (# E ( F p )) satisfies an Erd˝ os–Kac law with mean log log p and variance log log p . Erd˝ os–Pomerance (1985) ω ( φ ( n )) and Ω( φ ( n )) satisfy Erd˝ os– Kac laws with mean 2 ( log log n ) 2 and variance 1 1 3 ( log log n ) 3 . Ω( φ ( n )) is not additive, but is “ φ -additive”: if φ ( n ) = p r 1 1 · · · p r k k , then Ω( φ ( n )) = Ω( p r 1 1 ) + · · · + Ω( p r k k ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Other functions with Erd˝ os–Kac laws The paper of Erd˝ os–Kac establishes these normal-distribution laws for a large class of additive functions: if n = p r 1 1 · · · p r k k , then f ( n ) = f ( p r 1 1 ) + · · · + f ( p r k k ) . Examples of non-additive functions: Liu (2007) On GRH, ω (# E ( F p )) satisfies an Erd˝ os–Kac law with mean log log p and variance log log p . Erd˝ os–Pomerance (1985) ω ( φ ( n )) and Ω( φ ( n )) satisfy Erd˝ os– Kac laws with mean 2 ( log log n ) 2 and variance 1 1 3 ( log log n ) 3 . Ω( φ ( n )) is not additive, but is “ φ -additive”: if φ ( n ) = p r 1 1 · · · p r k k , then Ω( φ ( n )) = Ω( p r 1 1 ) + · · · + Ω( p r k k ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n A different question about the multiplicative group Notation reminder M n = Z × n , an abelian group with φ ( n ) elements. Question (Vukoslavcevic and Shparlinski, 2010) How many subgroups does M n have? Notation (used throughout this section of the talk) I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups (that is, subgroups not up to isomorphism). Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n A different question about the multiplicative group Notation reminder M n = Z × n , an abelian group with φ ( n ) elements. Question (Vukoslavcevic and Shparlinski, 2010) How many subgroups does M n have? Notation (used throughout this section of the talk) I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups (that is, subgroups not up to isomorphism). Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n A different question about the multiplicative group Notation reminder M n = Z × n , an abelian group with φ ( n ) elements. Question (Vukoslavcevic and Shparlinski, 2010) How many subgroups does M n have? Notation (used throughout this section of the talk) I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups (that is, subgroups not up to isomorphism). Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n A different question about the multiplicative group Notation reminder M n = Z × n , an abelian group with φ ( n ) elements. Question (Vukoslavcevic and Shparlinski, 2010) How many subgroups does M n have? Notation (used throughout this section of the talk) I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups (that is, subgroups not up to isomorphism). Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The number of subgroups has a similar property Getting used to the notation I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups. Every finite abelian group is the direct sum of its p -Sylow subgroups, so consequently: If G p ( n ) denotes the number of subgroups of the p -Sylow � � subgroup of M n , then G ( n ) = G p ( n ) = G p ( n ) . p | # M n p | φ ( n ) And similarly for I ( n ) . In particular, both I ( n ) and G ( n ) are “ φ -multiplicative” functions; so we might hope to get strong distributional information for the φ -additive functions log I ( n ) and log G ( n ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The number of subgroups has a similar property Getting used to the notation I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups. Every finite abelian group is the direct sum of its p -Sylow subgroups, so consequently: If G p ( n ) denotes the number of subgroups of the p -Sylow � � subgroup of M n , then G ( n ) = G p ( n ) = G p ( n ) . p | # M n p | φ ( n ) And similarly for I ( n ) . In particular, both I ( n ) and G ( n ) are “ φ -multiplicative” functions; so we might hope to get strong distributional information for the φ -additive functions log I ( n ) and log G ( n ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The number of subgroups has a similar property Getting used to the notation I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups. Every finite abelian group is the direct sum of its p -Sylow subgroups, so consequently: If G p ( n ) denotes the number of subgroups of the p -Sylow � � subgroup of M n , then G ( n ) = G p ( n ) = G p ( n ) . p | # M n p | φ ( n ) And similarly for I ( n ) . In particular, both I ( n ) and G ( n ) are “ φ -multiplicative” functions; so we might hope to get strong distributional information for the φ -additive functions log I ( n ) and log G ( n ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The number of subgroups has a similar property Getting used to the notation I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups. Every finite abelian group is the direct sum of its p -Sylow subgroups, so consequently: If G p ( n ) denotes the number of subgroups of the p -Sylow � � subgroup of M n , then G ( n ) = G p ( n ) = G p ( n ) . p | # M n p | φ ( n ) And similarly for I ( n ) . In particular, both I ( n ) and G ( n ) are “ φ -multiplicative” functions; so we might hope to get strong distributional information for the φ -additive functions log I ( n ) and log G ( n ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The number of subgroups has a similar property Getting used to the notation I ( n ) is the number of isomorphism classes of subgroups of M n . G ( n ) is the number of subsets of M n that are subgroups. Every finite abelian group is the direct sum of its p -Sylow subgroups, so consequently: If G p ( n ) denotes the number of subgroups of the p -Sylow � � subgroup of M n , then G ( n ) = G p ( n ) = G p ( n ) . p | # M n p | φ ( n ) And similarly for I ( n ) . In particular, both I ( n ) and G ( n ) are “ φ -multiplicative” functions; so we might hope to get strong distributional information for the φ -additive functions log I ( n ) and log G ( n ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws for the number of subgroups Theorem (M.–Troupe, to appear in the Journal of the Australian Mathematical Society) os–Kac law with mean log 2 2 ( log log n ) 2 log I ( n ) satisfies an Erd˝ and variance log 2 3 ( log log n ) 3 . How did we prove this? We showed that ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 , and then quoted Erd˝ os–Pomerance. Theorem (M.–Troupe) os–Kac law with mean A ( log log n ) 2 and log G ( n ) satisfies an Erd˝ variance C ( log log n ) 3 , for certain constants A and C . log 2 ≈ 0 . 34657 while A ≈ 0 . 72109 , so typically G ( n ) ≈ I ( n ) 2 . 08 . 2 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws for the number of subgroups Theorem (M.–Troupe, to appear in the Journal of the Australian Mathematical Society) os–Kac law with mean log 2 2 ( log log n ) 2 log I ( n ) satisfies an Erd˝ and variance log 2 3 ( log log n ) 3 . How did we prove this? We showed that ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 , and then quoted Erd˝ os–Pomerance. Theorem (M.–Troupe) os–Kac law with mean A ( log log n ) 2 and log G ( n ) satisfies an Erd˝ variance C ( log log n ) 3 , for certain constants A and C . log 2 ≈ 0 . 34657 while A ≈ 0 . 72109 , so typically G ( n ) ≈ I ( n ) 2 . 08 . 2 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws for the number of subgroups Theorem (M.–Troupe, to appear in the Journal of the Australian Mathematical Society) os–Kac law with mean log 2 2 ( log log n ) 2 log I ( n ) satisfies an Erd˝ and variance log 2 3 ( log log n ) 3 . How did we prove this? We showed that ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 , and then quoted Erd˝ os–Pomerance. Theorem (M.–Troupe) os–Kac law with mean A ( log log n ) 2 and log G ( n ) satisfies an Erd˝ variance C ( log log n ) 3 , for certain constants A and C . log 2 ≈ 0 . 34657 while A ≈ 0 . 72109 , so typically G ( n ) ≈ I ( n ) 2 . 08 . 2 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Erd˝ os–Kac laws for the number of subgroups Theorem (M.–Troupe, to appear in the Journal of the Australian Mathematical Society) os–Kac law with mean log 2 2 ( log log n ) 2 log I ( n ) satisfies an Erd˝ and variance log 2 3 ( log log n ) 3 . How did we prove this? We showed that ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 , and then quoted Erd˝ os–Pomerance. Theorem (M.–Troupe) os–Kac law with mean A ( log log n ) 2 and log G ( n ) satisfies an Erd˝ variance C ( log log n ) 3 , for certain constants A and C . log 2 ≈ 0 . 34657 while A ≈ 0 . 72109 , so typically G ( n ) ≈ I ( n ) 2 . 08 . 2 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n We had to look at these constants, so you do too Definition p 2 log p A 0 = 1 � ( p − 1 ) 3 ( p + 1 ) 4 p A = log 2 + A 0 ≈ 0 . 72109 2 p 3 ( p 4 − p 3 − p 3 − p − 1 )( log p ) 2 B = 1 � ( p − 1 ) 6 ( p + 1 ) 2 ( p 2 + p + 1 ) 4 p C = ( log 2 ) 2 + 2 A 0 log 2 + 4 A 2 0 + B ≈ 3 . 924 3 (The two sums are convergent sums over all primes p .) Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n How many subgroups can there be? Theorem (M.–Troupe) The order of magnitude of the maximal order of log I ( n ) is log n / log log n . More precisely, � log 2 log x 2 log x � � log log x � max log I ( n ) � π log log x . 5 3 n ≤ x Theorem (M.–Troupe) The order of magnitude of the maximal order of log G ( n ) is ( log n ) 2 / log log n . More precisely, ( log x ) 2 ( log x ) 2 1 � 1 � � log G ( n ) log log x . log log x � max 16 4 n ≤ x Consequence: G ( n ) can be superpolynomially large There are infinitely many integers n with G ( n ) > n 2018 ! . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n How many subgroups can there be? Theorem (M.–Troupe) The order of magnitude of the maximal order of log I ( n ) is log n / log log n . More precisely, � log 2 log x 2 log x � � log log x � max log I ( n ) � π log log x . 5 3 n ≤ x Theorem (M.–Troupe) The order of magnitude of the maximal order of log G ( n ) is ( log n ) 2 / log log n . More precisely, ( log x ) 2 ( log x ) 2 1 � 1 � � log G ( n ) log log x . log log x � max 16 4 n ≤ x Consequence: G ( n ) can be superpolynomially large There are infinitely many integers n with G ( n ) > n 2018 ! . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n How many subgroups can there be? Theorem (M.–Troupe) The order of magnitude of the maximal order of log I ( n ) is log n / log log n . More precisely, � log 2 log x 2 log x � � log log x � max log I ( n ) � π log log x . 5 3 n ≤ x Theorem (M.–Troupe) The order of magnitude of the maximal order of log G ( n ) is ( log n ) 2 / log log n . More precisely, ( log x ) 2 ( log x ) 2 1 � 1 � � log G ( n ) log log x . log log x � max 16 4 n ≤ x Consequence: G ( n ) can be superpolynomially large There are infinitely many integers n with G ( n ) > n 2018 ! . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Finite abelian groups and partitions Facts about finite abelian p -groups Every finite abelian group of size p m can be written uniquely as C p α = C p α 1 ⊕ C p α 2 ⊕ · · · ⊕ C p αℓ for some partition α = ( α 1 , α 2 , . . . , α ℓ ) of m (so α 1 ≥ α 2 ≥ · · · ≥ α ℓ ). So the number of isomorphism classes of subgroups of C p α is exactly the number of subpartitions β � α . . . . . . which is somewhere between 2 and 2 m inclusive. In other words: log # { subpartitions of α } is between log 2 and m log 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Finite abelian groups and partitions Facts about finite abelian p -groups Every finite abelian group of size p m can be written uniquely as C p α = C p α 1 ⊕ C p α 2 ⊕ · · · ⊕ C p αℓ for some partition α = ( α 1 , α 2 , . . . , α ℓ ) of m (so α 1 ≥ α 2 ≥ · · · ≥ α ℓ ). So the number of isomorphism classes of subgroups of C p α is exactly the number of subpartitions β � α . . . . . . which is somewhere between 2 and 2 m inclusive. In other words: log # { subpartitions of α } is between log 2 and m log 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Finite abelian groups and partitions Facts about finite abelian p -groups Every finite abelian group of size p m can be written uniquely as C p α = C p α 1 ⊕ C p α 2 ⊕ · · · ⊕ C p αℓ for some partition α = ( α 1 , α 2 , . . . , α ℓ ) of m (so α 1 ≥ α 2 ≥ · · · ≥ α ℓ ). So the number of isomorphism classes of subgroups of C p α is exactly the number of subpartitions β � α . . . . . . which is somewhere between 2 and 2 m inclusive. In other words: log # { subpartitions of α } is between log 2 and m log 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Finite abelian groups and partitions Facts about finite abelian p -groups Every finite abelian group of size p m can be written uniquely as C p α = C p α 1 ⊕ C p α 2 ⊕ · · · ⊕ C p αℓ for some partition α = ( α 1 , α 2 , . . . , α ℓ ) of m (so α 1 ≥ α 2 ≥ · · · ≥ α ℓ ). So the number of isomorphism classes of subgroups of C p α is exactly the number of subpartitions β � α . . . . . . which is somewhere between 2 and 2 m inclusive. In other words: log # { subpartitions of α } is between log 2 and m log 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Application to distribution of I ( n ) I ( n ) is the number of isomorphism classes of subgroups of M n More notation p | φ ( n ) p m ( p ) , so that M n ∼ Let φ ( n ) = � = � p | φ ( n ) C p α ( p ) for some partitions α ( p ) of m ( p ) . Then log I ( n ) = � p | φ ( n ) log # { subpartitions of α p } and hence � � log 2 ≤ log I ( n ) ≤ m ( p ) log 2 p | φ ( n ) p | φ ( n ) ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 Upper bound seems very wasteful, yet still good enough! “Anatomy of integers” techniques show: most primes dividing φ ( n ) do so only once. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Application to distribution of I ( n ) I ( n ) is the number of isomorphism classes of subgroups of M n More notation p | φ ( n ) p m ( p ) , so that M n ∼ Let φ ( n ) = � = � p | φ ( n ) C p α ( p ) for some partitions α ( p ) of m ( p ) . Then log I ( n ) = � p | φ ( n ) log # { subpartitions of α p } and hence � � log 2 ≤ log I ( n ) ≤ m ( p ) log 2 p | φ ( n ) p | φ ( n ) ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 Upper bound seems very wasteful, yet still good enough! “Anatomy of integers” techniques show: most primes dividing φ ( n ) do so only once. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Application to distribution of I ( n ) I ( n ) is the number of isomorphism classes of subgroups of M n More notation p | φ ( n ) p m ( p ) , so that M n ∼ Let φ ( n ) = � = � p | φ ( n ) C p α ( p ) for some partitions α ( p ) of m ( p ) . Then log I ( n ) = � p | φ ( n ) log # { subpartitions of α p } and hence � � log 2 ≤ log I ( n ) ≤ m ( p ) log 2 p | φ ( n ) p | φ ( n ) ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 Upper bound seems very wasteful, yet still good enough! “Anatomy of integers” techniques show: most primes dividing φ ( n ) do so only once. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Application to distribution of I ( n ) I ( n ) is the number of isomorphism classes of subgroups of M n More notation p | φ ( n ) p m ( p ) , so that M n ∼ Let φ ( n ) = � = � p | φ ( n ) C p α ( p ) for some partitions α ( p ) of m ( p ) . Then log I ( n ) = � p | φ ( n ) log # { subpartitions of α p } and hence � � log 2 ≤ log I ( n ) ≤ m ( p ) log 2 p | φ ( n ) p | φ ( n ) ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 Upper bound seems very wasteful, yet still good enough! “Anatomy of integers” techniques show: most primes dividing φ ( n ) do so only once. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Application to distribution of I ( n ) I ( n ) is the number of isomorphism classes of subgroups of M n More notation p | φ ( n ) p m ( p ) , so that M n ∼ Let φ ( n ) = � = � p | φ ( n ) C p α ( p ) for some partitions α ( p ) of m ( p ) . Then log I ( n ) = � p | φ ( n ) log # { subpartitions of α p } and hence � � log 2 ≤ log I ( n ) ≤ m ( p ) log 2 p | φ ( n ) p | φ ( n ) ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 Upper bound seems very wasteful, yet still good enough! “Anatomy of integers” techniques show: most primes dividing φ ( n ) do so only once. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Application to distribution of I ( n ) I ( n ) is the number of isomorphism classes of subgroups of M n More notation p | φ ( n ) p m ( p ) , so that M n ∼ Let φ ( n ) = � = � p | φ ( n ) C p α ( p ) for some partitions α ( p ) of m ( p ) . Then log I ( n ) = � p | φ ( n ) log # { subpartitions of α p } and hence � � log 2 ≤ log I ( n ) ≤ m ( p ) log 2 p | φ ( n ) p | φ ( n ) ω ( φ ( n )) log 2 ≤ log I ( n ) ≤ Ω( φ ( n )) log 2 Upper bound seems very wasteful, yet still good enough! “Anatomy of integers” techniques show: most primes dividing φ ( n ) do so only once. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n How many subgroups of each shape? Notation: α = ( α 1 , . . . , α ℓ ) , C p α = C p α 1 ⊕ · · · ⊕ C p αℓ Definition Given a subpartition β of α and a prime p , define N p ( α , β ) to be the number of subgroups inside C p α that are isomorphic to C p β . Some classical exact formula (don’t read it) Let a = ( a 1 , a 2 , . . . , a α 1 ) and b = ( b 1 , b 2 , . . . , b β 1 ) be the conjugate partitions to α and β , respectively. Then α 1 � a j − b j + 1 � � p ( a j − b j ) b j + 1 N p ( α , β ) = , b j − b j + 1 p j = 1 p k − ℓ + j − 1 p = � ℓ � k � where is the Gaussian binomial coefficient. ℓ j = 1 p j − 1 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n How many subgroups of each shape? Notation: α = ( α 1 , . . . , α ℓ ) , C p α = C p α 1 ⊕ · · · ⊕ C p αℓ Definition Given a subpartition β of α and a prime p , define N p ( α , β ) to be the number of subgroups inside C p α that are isomorphic to C p β . Some classical exact formula (don’t read it) Let a = ( a 1 , a 2 , . . . , a α 1 ) and b = ( b 1 , b 2 , . . . , b β 1 ) be the conjugate partitions to α and β , respectively. Then α 1 � a j − b j + 1 � � p ( a j − b j ) b j + 1 N p ( α , β ) = , b j − b j + 1 p j = 1 p k − ℓ + j − 1 p = � ℓ � k � where is the Gaussian binomial coefficient. ℓ j = 1 p j − 1 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The difference between algebra and analysis α 1 � a j − b j + 1 � � p ( a j − b j ) b j + 1 N p ( α , β ) = b j − b j + 1 p j = 1 is the number of subgroups inside C p α isomorphic to C p β . It turns out that each factor is about p ( a j − b j ) b j , which is j / 4 when b j = a j / 2 , and is way smaller for maximally p a 2 noncentral values of b j . So the total number of subgroups inside C p α is dominated by this special β = “ 1 2 α ”. Lemma For any prime p and any partition α , α 1 log # { subgroups of C p α } = log p � a 2 j + O ( α 1 log p ) . 4 j = 1 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The difference between algebra and analysis α 1 � a j − b j + 1 � � p ( a j − b j ) b j + 1 N p ( α , β ) = b j − b j + 1 p j = 1 is the number of subgroups inside C p α isomorphic to C p β . It turns out that each factor is about p ( a j − b j ) b j , which is j / 4 when b j = a j / 2 , and is way smaller for maximally p a 2 noncentral values of b j . So the total number of subgroups inside C p α is dominated by this special β = “ 1 2 α ”. Lemma For any prime p and any partition α , α 1 log # { subgroups of C p α } = log p � a 2 j + O ( α 1 log p ) . 4 j = 1 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The difference between algebra and analysis α 1 � a j − b j + 1 � � p ( a j − b j ) b j + 1 N p ( α , β ) = b j − b j + 1 p j = 1 is the number of subgroups inside C p α isomorphic to C p β . It turns out that each factor is about p ( a j − b j ) b j , which is j / 4 when b j = a j / 2 , and is way smaller for maximally p a 2 noncentral values of b j . So the total number of subgroups inside C p α is dominated by this special β = “ 1 2 α ”. Lemma For any prime p and any partition α , α 1 log # { subgroups of C p α } = log p � a 2 j + O ( α 1 log p ) . 4 j = 1 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The difference between algebra and analysis α 1 � a j − b j + 1 � � p ( a j − b j ) b j + 1 N p ( α , β ) = b j − b j + 1 p j = 1 is the number of subgroups inside C p α isomorphic to C p β . It turns out that each factor is about p ( a j − b j ) b j , which is j / 4 when b j = a j / 2 , and is way smaller for maximally p a 2 noncentral values of b j . So the total number of subgroups inside C p α is dominated by this special β = “ 1 2 α ”. Lemma For any prime p and any partition α , α 1 log # { subgroups of C p α } = log p � a 2 j + O ( α 1 log p ) . 4 j = 1 Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n If M n ∼ = � p | φ ( n ) C p α ( p ) , then which partition is α ( p ) ? Notation Let ω q ( n ) denote the number of distinct prime factors of n that are congruent to 1 (mod q ) . Answer (exact for odd squarefree n , up to O ( 1 ) in general) � � α ( p ) is the conjugate partition to ω p ( n ) , ω p 2 ( n ) , . . . . Lemma “ ∞ ” log G p ( n ) ≈ log p ω p j ( n ) 2 for any prime p dividing φ ( n ) . � 4 j = 1 Moreover, if p | φ ( n ) and p 2 ∤ φ ( n ) , then log G p ( n ) = log 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n If M n ∼ = � p | φ ( n ) C p α ( p ) , then which partition is α ( p ) ? Notation Let ω q ( n ) denote the number of distinct prime factors of n that are congruent to 1 (mod q ) . Answer (exact for odd squarefree n , up to O ( 1 ) in general) � � α ( p ) is the conjugate partition to ω p ( n ) , ω p 2 ( n ) , . . . . Lemma “ ∞ ” log G p ( n ) ≈ log p ω p j ( n ) 2 for any prime p dividing φ ( n ) . � 4 j = 1 Moreover, if p | φ ( n ) and p 2 ∤ φ ( n ) , then log G p ( n ) = log 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n If M n ∼ = � p | φ ( n ) C p α ( p ) , then which partition is α ( p ) ? Notation Let ω q ( n ) denote the number of distinct prime factors of n that are congruent to 1 (mod q ) . Answer (exact for odd squarefree n , up to O ( 1 ) in general) � � α ( p ) is the conjugate partition to ω p ( n ) , ω p 2 ( n ) , . . . . Lemma “ ∞ ” log G p ( n ) ≈ log p ω p j ( n ) 2 for any prime p dividing φ ( n ) . � 4 j = 1 Moreover, if p | φ ( n ) and p 2 ∤ φ ( n ) , then log G p ( n ) = log 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n If M n ∼ = � p | φ ( n ) C p α ( p ) , then which partition is α ( p ) ? Notation Let ω q ( n ) denote the number of distinct prime factors of n that are congruent to 1 (mod q ) . Answer (exact for odd squarefree n , up to O ( 1 ) in general) � � α ( p ) is the conjugate partition to ω p ( n ) , ω p 2 ( n ) , . . . . Lemma “ ∞ ” log G p ( n ) ≈ log p ω p j ( n ) 2 for any prime p dividing φ ( n ) . � 4 j = 1 Moreover, if p | φ ( n ) and p 2 ∤ φ ( n ) , then log G p ( n ) = log 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Sum the previous lemma over all primes “ ∞ ” log p � � � � ω p j ( n ) 2 . log G ( n ) = log G p ( n ) ≈ log 2 + 4 p | φ ( n ) p | φ ( n ) p 2 | φ ( n ) j = 1 p 2 ∤ φ ( n ) For most integers n , it’s acceptable to extend both sums over all primes dividing φ ( n ) (the last sum should be suitably truncated): log G ( n ) ≈ log 2 · ω ( φ ( n )) + 1 ω p r ( n ) 2 log p . � 4 p r Each function here has a known normal order; plugging in gives 2 � log log n � log G ( n ) ≈ log 2 · 1 2 ( log log n ) 2 + 1 � log p φ ( p r ) 4 p r for almost all integers n . And the right-hand side is A ( log log n ) 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Sum the previous lemma over all primes “ ∞ ” log p � � � � ω p j ( n ) 2 . log G ( n ) = log G p ( n ) ≈ log 2 + 4 p | φ ( n ) p | φ ( n ) p 2 | φ ( n ) j = 1 p 2 ∤ φ ( n ) For most integers n , it’s acceptable to extend both sums over all primes dividing φ ( n ) (the last sum should be suitably truncated): log G ( n ) ≈ log 2 · ω ( φ ( n )) + 1 ω p r ( n ) 2 log p . � 4 p r Each function here has a known normal order; plugging in gives 2 � log log n � log G ( n ) ≈ log 2 · 1 2 ( log log n ) 2 + 1 � log p φ ( p r ) 4 p r for almost all integers n . And the right-hand side is A ( log log n ) 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Sum the previous lemma over all primes “ ∞ ” log p � � � � ω p j ( n ) 2 . log G ( n ) = log G p ( n ) ≈ log 2 + 4 p | φ ( n ) p | φ ( n ) p 2 | φ ( n ) j = 1 p 2 ∤ φ ( n ) For most integers n , it’s acceptable to extend both sums over all primes dividing φ ( n ) (the last sum should be suitably truncated): log G ( n ) ≈ log 2 · ω ( φ ( n )) + 1 ω p r ( n ) 2 log p . � 4 p r Each function here has a known normal order; plugging in gives 2 � log log n � log G ( n ) ≈ log 2 · 1 2 ( log log n ) 2 + 1 � log p φ ( p r ) 4 p r for almost all integers n . And the right-hand side is A ( log log n ) 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Sum the previous lemma over all primes “ ∞ ” log p � � � � ω p j ( n ) 2 . log G ( n ) = log G p ( n ) ≈ log 2 + 4 p | φ ( n ) p | φ ( n ) p 2 | φ ( n ) j = 1 p 2 ∤ φ ( n ) For most integers n , it’s acceptable to extend both sums over all primes dividing φ ( n ) (the last sum should be suitably truncated): log G ( n ) ≈ log 2 · ω ( φ ( n )) + 1 ω p r ( n ) 2 log p . � 4 p r Each function here has a known normal order; plugging in gives 2 � log log n � log G ( n ) ≈ log 2 · 1 2 ( log log n ) 2 + 1 � log p φ ( p r ) 4 p r for almost all integers n . And the right-hand side is A ( log log n ) 2 . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Final sketch Getting beyond the normal order to an Erd˝ os–Kac law requires computing all of the central moments of this approximation to log G ( n ) . The correlations among the additive functions ω q ( n ) , and their correlations with ω ( φ ( n )) , become important. “Sieving and the Erd˝ os–Kac theorem” (2007) To compute the moments, we rely on a technique of Granville and Soundararajan to reduce the complexity of identifying the main terms of these moments. Generalizing our method Part of log G ( n ) is well approximated by a sum of squares of additive functions. Troupe and I (just submitted!) can obtain an Erd˝ os–Kac law for any fixed nonnegative polynomial evaluated at values of appropriate additive functions—for example, Erd˝ os–Kac laws for products of additive functions. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Final sketch Getting beyond the normal order to an Erd˝ os–Kac law requires computing all of the central moments of this approximation to log G ( n ) . The correlations among the additive functions ω q ( n ) , and their correlations with ω ( φ ( n )) , become important. “Sieving and the Erd˝ os–Kac theorem” (2007) To compute the moments, we rely on a technique of Granville and Soundararajan to reduce the complexity of identifying the main terms of these moments. Generalizing our method Part of log G ( n ) is well approximated by a sum of squares of additive functions. Troupe and I (just submitted!) can obtain an Erd˝ os–Kac law for any fixed nonnegative polynomial evaluated at values of appropriate additive functions—for example, Erd˝ os–Kac laws for products of additive functions. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Final sketch Getting beyond the normal order to an Erd˝ os–Kac law requires computing all of the central moments of this approximation to log G ( n ) . The correlations among the additive functions ω q ( n ) , and their correlations with ω ( φ ( n )) , become important. “Sieving and the Erd˝ os–Kac theorem” (2007) To compute the moments, we rely on a technique of Granville and Soundararajan to reduce the complexity of identifying the main terms of these moments. Generalizing our method Part of log G ( n ) is well approximated by a sum of squares of additive functions. Troupe and I (just submitted!) can obtain an Erd˝ os–Kac law for any fixed nonnegative polynomial evaluated at values of appropriate additive functions—for example, Erd˝ os–Kac laws for products of additive functions. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Final sketch Getting beyond the normal order to an Erd˝ os–Kac law requires computing all of the central moments of this approximation to log G ( n ) . The correlations among the additive functions ω q ( n ) , and their correlations with ω ( φ ( n )) , become important. “Sieving and the Erd˝ os–Kac theorem” (2007) To compute the moments, we rely on a technique of Granville and Soundararajan to reduce the complexity of identifying the main terms of these moments. Generalizing our method Part of log G ( n ) is well approximated by a sum of squares of additive functions. Troupe and I (just submitted!) can obtain an Erd˝ os–Kac law for any fixed nonnegative polynomial evaluated at values of appropriate additive functions—for example, Erd˝ os–Kac laws for products of additive functions. Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Prohibited prime factors: related problems Let q be a fixed odd prime. Definition The q -Sylow subgroup of a finite abelian group G is the largest subgroup of G whose cardinality is a power of q . Question (Colin Weir, 2017) If we fix our favorite prime q and our favorite group G , how often will we get G as the exact q -Sylow subgroup of M n ? Definition S q , G ( x ) = # { n ≤ x : the q -Sylow subgroup of M n equals G } Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Prohibited prime factors: related problems Let q be a fixed odd prime. Definition The q -Sylow subgroup of a finite abelian group G is the largest subgroup of G whose cardinality is a power of q . Question (Colin Weir, 2017) If we fix our favorite prime q and our favorite group G , how often will we get G as the exact q -Sylow subgroup of M n ? Definition S q , G ( x ) = # { n ≤ x : the q -Sylow subgroup of M n equals G } Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Prohibited prime factors: related problems Let q be a fixed odd prime. Definition The q -Sylow subgroup of a finite abelian group G is the largest subgroup of G whose cardinality is a power of q . Question (Colin Weir, 2017) If we fix our favorite prime q and our favorite group G , how often will we get G as the exact q -Sylow subgroup of M n ? Definition S q , G ( x ) = # { n ≤ x : the q -Sylow subgroup of M n equals G } Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The trivial example Another translation of algebra to number theory { n : the q -Sylow subgroup of M n is trivial } = { n : q does not divide # M n = φ ( n ) } and q 2 ∤ n } . � � = { n : p | n = ⇒ p �≡ 1 (mod q ) Example (when G = { 1 } is trivial) S q , { 1 } ( x ) = # { n ≤ x : the q -Sylow subgroup of M n is trivial } and q 2 ∤ n } : � � = # { n : p | n = ⇒ p �≡ 1 (mod q ) we prohibit prime factors from a set of relative density 1 / ( q − 1 ) . General philosophy Prohibiting prime divisors from a set of primes of relative density δ divides the counting function by a factor of ( log x ) δ . Correspondingly, we expect S q , { 1 } ( x ) ≍ x / ( log x ) 1 / ( q − 1 ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The trivial example Another translation of algebra to number theory { n : the q -Sylow subgroup of M n is trivial } = { n : q does not divide # M n = φ ( n ) } and q 2 ∤ n } . � � = { n : p | n = ⇒ p �≡ 1 (mod q ) Example (when G = { 1 } is trivial) S q , { 1 } ( x ) = # { n ≤ x : the q -Sylow subgroup of M n is trivial } and q 2 ∤ n } : � � = # { n : p | n = ⇒ p �≡ 1 (mod q ) we prohibit prime factors from a set of relative density 1 / ( q − 1 ) . General philosophy Prohibiting prime divisors from a set of primes of relative density δ divides the counting function by a factor of ( log x ) δ . Correspondingly, we expect S q , { 1 } ( x ) ≍ x / ( log x ) 1 / ( q − 1 ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The trivial example Another translation of algebra to number theory { n : the q -Sylow subgroup of M n is trivial } = { n : q does not divide # M n = φ ( n ) } and q 2 ∤ n } . � � = { n : p | n = ⇒ p �≡ 1 (mod q ) Example (when G = { 1 } is trivial) S q , { 1 } ( x ) = # { n ≤ x : the q -Sylow subgroup of M n is trivial } and q 2 ∤ n } : � � = # { n : p | n = ⇒ p �≡ 1 (mod q ) we prohibit prime factors from a set of relative density 1 / ( q − 1 ) . General philosophy Prohibiting prime divisors from a set of primes of relative density δ divides the counting function by a factor of ( log x ) δ . Correspondingly, we expect S q , { 1 } ( x ) ≍ x / ( log x ) 1 / ( q − 1 ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The trivial example Another translation of algebra to number theory { n : the q -Sylow subgroup of M n is trivial } = { n : q does not divide # M n = φ ( n ) } and q 2 ∤ n } . � � = { n : p | n = ⇒ p �≡ 1 (mod q ) Example (when G = { 1 } is trivial) S q , { 1 } ( x ) = # { n ≤ x : the q -Sylow subgroup of M n is trivial } and q 2 ∤ n } : � � = # { n : p | n = ⇒ p �≡ 1 (mod q ) we prohibit prime factors from a set of relative density 1 / ( q − 1 ) . General philosophy Prohibiting prime divisors from a set of primes of relative density δ divides the counting function by a factor of ( log x ) δ . Correspondingly, we expect S q , { 1 } ( x ) ≍ x / ( log x ) 1 / ( q − 1 ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The trivial example Another translation of algebra to number theory { n : the q -Sylow subgroup of M n is trivial } = { n : q does not divide # M n = φ ( n ) } and q 2 ∤ n } . � � = { n : p | n = ⇒ p �≡ 1 (mod q ) Example (when G = { 1 } is trivial) S q , { 1 } ( x ) = # { n ≤ x : the q -Sylow subgroup of M n is trivial } and q 2 ∤ n } : � � = # { n : p | n = ⇒ p �≡ 1 (mod q ) we prohibit prime factors from a set of relative density 1 / ( q − 1 ) . General philosophy Prohibiting prime divisors from a set of primes of relative density δ divides the counting function by a factor of ( log x ) δ . Correspondingly, we expect S q , { 1 } ( x ) ≍ x / ( log x ) 1 / ( q − 1 ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The trivial example Another translation of algebra to number theory { n : the q -Sylow subgroup of M n is trivial } = { n : q does not divide # M n = φ ( n ) } and q 2 ∤ n } . � � = { n : p | n = ⇒ p �≡ 1 (mod q ) Example (when G = { 1 } is trivial) S q , { 1 } ( x ) = # { n ≤ x : the q -Sylow subgroup of M n is trivial } and q 2 ∤ n } : � � = # { n : p | n = ⇒ p �≡ 1 (mod q ) we prohibit prime factors from a set of relative density 1 / ( q − 1 ) . General philosophy Prohibiting prime divisors from a set of primes of relative density δ divides the counting function by a factor of ( log x ) δ . Correspondingly, we expect S q , { 1 } ( x ) ≍ x / ( log x ) 1 / ( q − 1 ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The trivial example Another translation of algebra to number theory { n : the q -Sylow subgroup of M n is trivial } = { n : q does not divide # M n = φ ( n ) } and q 2 ∤ n } . � � = { n : p | n = ⇒ p �≡ 1 (mod q ) Example (when G = { 1 } is trivial) S q , { 1 } ( x ) = # { n ≤ x : the q -Sylow subgroup of M n is trivial } and q 2 ∤ n } : � � = # { n : p | n = ⇒ p �≡ 1 (mod q ) we prohibit prime factors from a set of relative density 1 / ( q − 1 ) . General philosophy Prohibiting prime divisors from a set of primes of relative density δ divides the counting function by a factor of ( log x ) δ . Correspondingly, we expect S q , { 1 } ( x ) ≍ x / ( log x ) 1 / ( q − 1 ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The trivial example Another translation of algebra to number theory { n : the q -Sylow subgroup of M n is trivial } = { n : q does not divide # M n = φ ( n ) } and q 2 ∤ n } . � � = { n : p | n = ⇒ p �≡ 1 (mod q ) Example (when G = { 1 } is trivial) S q , { 1 } ( x ) = # { n ≤ x : the q -Sylow subgroup of M n is trivial } and q 2 ∤ n } : � � = # { n : p | n = ⇒ p �≡ 1 (mod q ) we prohibit prime factors from a set of relative density 1 / ( q − 1 ) . General philosophy Prohibiting prime divisors from a set of primes of relative density δ divides the counting function by a factor of ( log x ) δ . Correspondingly, we expect S q , { 1 } ( x ) ≍ x / ( log x ) 1 / ( q − 1 ) . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Analytic number theorists can do this Theorems of this type go back to Landau; today it would be deemed a standard application of the Selberg–Delange method; this particular application (other than the extra q 2 ∤ n ) was carried out by Ford/Luca/Moree (Math Comp., 2014). Theorem C q � � 1 − 1 x S q , { 1 } ∼ ( log x ) 1 / ( q − 1 ) ; 1 q 2 Γ( 1 − q − 1 ) here � − 1 / ( q − 1 ) � − 1 / k p �� � 1 − 1 � � � 1 − 1 C q = L ( 1 , χ ) , q p k p χ � = χ 0 p �≡ 0 , 1 (mod q ) where k p is the multiplicative order of p modulo q . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Analytic number theorists can do this Theorems of this type go back to Landau; today it would be deemed a standard application of the Selberg–Delange method; this particular application (other than the extra q 2 ∤ n ) was carried out by Ford/Luca/Moree (Math Comp., 2014). Theorem C q � � 1 − 1 x S q , { 1 } ∼ ( log x ) 1 / ( q − 1 ) ; 1 q 2 Γ( 1 − q − 1 ) here � − 1 / ( q − 1 ) � − 1 / k p �� � 1 − 1 � � � 1 − 1 C q = L ( 1 , χ ) , q p k p χ � = χ 0 p �≡ 0 , 1 (mod q ) where k p is the multiplicative order of p modulo q . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n Analytic number theorists can do this Theorems of this type go back to Landau; today it would be deemed a standard application of the Selberg–Delange method; this particular application (other than the extra q 2 ∤ n ) was carried out by Ford/Luca/Moree (Math Comp., 2014). Theorem C q � � 1 − 1 x S q , { 1 } ∼ ( log x ) 1 / ( q − 1 ) ; 1 q 2 Γ( 1 − q − 1 ) here � − 1 / ( q − 1 ) � − 1 / k p �� � 1 − 1 � � � 1 − 1 C q = L ( 1 , χ ) , q p k p χ � = χ 0 p �≡ 0 , 1 (mod q ) where k p is the multiplicative order of p modulo q . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The general case Suppose now that G = C q α : Typically this q -Sylow subgroup of M n arises from a single prime p | n with q α � ( p − 1 ) , and the other factor n p of the form discussed in the trivial example. For each possible p , we thus get a contribution of p ) 1 / ( q − 1 ) , or more simply, ≍ x p / ( log x ) 1 / ( q − 1 ) . ≍ x p / ( log x Summing over possible p yields ≍ log log x x / ( log x ) 1 / ( q − 1 ) . q α (Need to tread more carefully; we found a nice argument using partial summation, asymptotics for hypergeometric functions.) In general, when G = C q α 1 ⊕ · · · ⊕ C q αℓ : Identify a prime p | n with q α ℓ � ( p − 1 ) ; the cofactor n p will be an example with G = C q α 1 ⊕ · · · ⊕ C q αℓ − 1 ; then recurse. . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The general case Suppose now that G = C q α : Typically this q -Sylow subgroup of M n arises from a single prime p | n with q α � ( p − 1 ) , and the other factor n p of the form discussed in the trivial example. For each possible p , we thus get a contribution of p ) 1 / ( q − 1 ) , or more simply, ≍ x p / ( log x ) 1 / ( q − 1 ) . ≍ x p / ( log x Summing over possible p yields ≍ log log x x / ( log x ) 1 / ( q − 1 ) . q α (Need to tread more carefully; we found a nice argument using partial summation, asymptotics for hypergeometric functions.) In general, when G = C q α 1 ⊕ · · · ⊕ C q αℓ : Identify a prime p | n with q α ℓ � ( p − 1 ) ; the cofactor n p will be an example with G = C q α 1 ⊕ · · · ⊕ C q αℓ − 1 ; then recurse. . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The general case Suppose now that G = C q α : Typically this q -Sylow subgroup of M n arises from a single prime p | n with q α � ( p − 1 ) , and the other factor n p of the form discussed in the trivial example. For each possible p , we thus get a contribution of p ) 1 / ( q − 1 ) , or more simply, ≍ x p / ( log x ) 1 / ( q − 1 ) . ≍ x p / ( log x Summing over possible p yields ≍ log log x x / ( log x ) 1 / ( q − 1 ) . q α (Need to tread more carefully; we found a nice argument using partial summation, asymptotics for hypergeometric functions.) In general, when G = C q α 1 ⊕ · · · ⊕ C q αℓ : Identify a prime p | n with q α ℓ � ( p − 1 ) ; the cofactor n p will be an example with G = C q α 1 ⊕ · · · ⊕ C q αℓ − 1 ; then recurse. . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The general case Suppose now that G = C q α : Typically this q -Sylow subgroup of M n arises from a single prime p | n with q α � ( p − 1 ) , and the other factor n p of the form discussed in the trivial example. For each possible p , we thus get a contribution of p ) 1 / ( q − 1 ) , or more simply, ≍ x p / ( log x ) 1 / ( q − 1 ) . ≍ x p / ( log x Summing over possible p yields ≍ log log x x / ( log x ) 1 / ( q − 1 ) . q α (Need to tread more carefully; we found a nice argument using partial summation, asymptotics for hypergeometric functions.) In general, when G = C q α 1 ⊕ · · · ⊕ C q αℓ : Identify a prime p | n with q α ℓ � ( p − 1 ) ; the cofactor n p will be an example with G = C q α 1 ⊕ · · · ⊕ C q αℓ − 1 ; then recurse. . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The general case Suppose now that G = C q α : Typically this q -Sylow subgroup of M n arises from a single prime p | n with q α � ( p − 1 ) , and the other factor n p of the form discussed in the trivial example. For each possible p , we thus get a contribution of p ) 1 / ( q − 1 ) , or more simply, ≍ x p / ( log x ) 1 / ( q − 1 ) . ≍ x p / ( log x Summing over possible p yields ≍ log log x x / ( log x ) 1 / ( q − 1 ) . q α (Need to tread more carefully; we found a nice argument using partial summation, asymptotics for hypergeometric functions.) In general, when G = C q α 1 ⊕ · · · ⊕ C q αℓ : Identify a prime p | n with q α ℓ � ( p − 1 ) ; the cofactor n p will be an example with G = C q α 1 ⊕ · · · ⊕ C q αℓ − 1 ; then recurse. . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The general case Suppose now that G = C q α : Typically this q -Sylow subgroup of M n arises from a single prime p | n with q α � ( p − 1 ) , and the other factor n p of the form discussed in the trivial example. For each possible p , we thus get a contribution of p ) 1 / ( q − 1 ) , or more simply, ≍ x p / ( log x ) 1 / ( q − 1 ) . ≍ x p / ( log x Summing over possible p yields ≍ log log x x / ( log x ) 1 / ( q − 1 ) . q α (Need to tread more carefully; we found a nice argument using partial summation, asymptotics for hypergeometric functions.) In general, when G = C q α 1 ⊕ · · · ⊕ C q αℓ : Identify a prime p | n with q α ℓ � ( p − 1 ) ; the cofactor n p will be an example with G = C q α 1 ⊕ · · · ⊕ C q αℓ − 1 ; then recurse. . . . Statistics of the multiplicative group Greg Martin

The multiplicative group M n Counting subgroups of M n Prescribed q -Sylow subgroups of M n Least invariant factor of M n The general case Suppose now that G = C q α : Typically this q -Sylow subgroup of M n arises from a single prime p | n with q α � ( p − 1 ) , and the other factor n p of the form discussed in the trivial example. For each possible p , we thus get a contribution of p ) 1 / ( q − 1 ) , or more simply, ≍ x p / ( log x ) 1 / ( q − 1 ) . ≍ x p / ( log x Summing over possible p yields ≍ log log x x / ( log x ) 1 / ( q − 1 ) . q α (Need to tread more carefully; we found a nice argument using partial summation, asymptotics for hypergeometric functions.) In general, when G = C q α 1 ⊕ · · · ⊕ C q αℓ : Identify a prime p | n with q α ℓ � ( p − 1 ) ; the cofactor n p will be an example with G = C q α 1 ⊕ · · · ⊕ C q αℓ − 1 ; then recurse. . . . Statistics of the multiplicative group Greg Martin