DRAFT Group Isomorphism is tied up in knots. James B. Wilson - PowerPoint PPT Presentation

DRAFT Group Isomorphism is tied up in knots. James B. Wilson Colorado State University 1

2 Isomorphism problems in algebra today. DRAFT ∞ PresGrpIso O ( c n 2 / 3 ) BBGrpIso # ABEL ModIso PcGrpIso MatGrpIso PermGrpIso RingIso LieIso O ( c n 1 / 4 ln 2 n ) SemigrpIso GraphIso O ( c ln 2 n ) QuasigroupIso CayleyGroupIso n is bit-wise input size, e.g. graphs on v vertices has n ∈ Θ( v 2 ).

3 Early history of group isomorphism DRAFT ? =

4 the second weaved through the Theorem (Dehn 1909). DRAFT knot. The closed knot tied under-over Bummer: G 1 ∼ cannot be deformed continu- = G 2 . ously to the closed knot tied The obvious iso- But wait! over-under. morphism is orientation revers- ing. Proof. A continuous defor- There are infinitely many iso- mation of one knot K 1 to an- morphisms to check. Instead, other K 2 will make an orien- compute a finite generating set tation preserving isomorphism of the automorphism group. of the two knot groups G i = { [ S 1 → R 3 \ K i ] } . These all preserve orientation. All isomorphisms between G 1 The knot groups are gen- and G 2 are orientation revers- erated by two loops: one ing. � wrapped over a single string,

5 DRAFT Moral: Group isomorphism is a powerful calculation capable of describing huge diversity between objects in a humble set of generators. The Group Isomorphism Problem (Dehn 1911). Is this calculation actually possible?

6 Let H = � Y | S � �∼ Adian 1955, Rabin 1957. = 1. Set DRAFT Group isomorphism for groups G = T ( H ∗ G 0 , w ) . given as If w ≡ 1 in G 0 then G ∼ = 1; else, 1 � = H ≤ H ∗ G 0 ≤ G . G = � x 1 , . . . , x n | r 1 , . . . , r m � So w ≡ 1 in G 0 iff G ∼ = 1. We is undecidable. 1 cannot decide this. Also, for any group K , Proof. Novikov ‘52/Boone ‘54 K ∗ G ∼ = K ⇔ G ∼ = 1 . create groups G 0 = � X | R � and So ( K, K ∗ G ) is a pair for which a word w such that w ≡ 1 in group isomorphism is undecideable. G 0 is undecidable. � Rabin: for every such w there is a group T ( G, w ) where w ≡ 1 in G implies T ∼ = 1; otherwise G ֒ → T . 1 IsIso ( G, H ) modeled as f : N → { 0 , 1 } is non-recursive. Recursive is rare – their are only countably many programs; yet, 2 N is uncountable.

7 Ouch. Cannot decide if Outside of algebra. Can- DRAFT groups are finite, abelian, not decide if spaces are homo- solvable, or indecomposable. topic. Proof. Consider Ellenberg- Proof. Fix a property P that MacLane spaces. � transfers to all subgroups (e.g. trivial, finite, abelian solvable, Reality check. etc.). Let H and K be groups, • Groups you find come with H with P and K without. Set more than � X | R � . • Gromov style “random” G = H ∗ T ( K ∗ G 0 , w ) . groups � X | R � have a solv- Cannot decide P for G . able word problem (they If H be directly indecompos- are hyperbolic.) able and G a group that we • Rabin. Isomorphism types cannot decide is trivial. Then are recursively enumerable. cannot decide if H × G is inde- composable. �

8 DRAFT Moral: Before deciding, ask the actual question. Single-set isomorphism: Given two group multiplications on the same set , how hard is it to solve group isomorphism?

9 Def. The generator degree of � DRAFT a group G is the cardinal: 2 Open Problem. d ( G ) = min {| X | : G = � X �} . Decide single-set isomorphism in time better than n O (log n ) . Fact. If d ( G ) � = | G | then 2 d ( G ) ≤ | G | ≤ ℵ 0 . Who opened this problem? Dehn 1911, Felsch-Neub¨ user Fact. For sets of size n ‘68, Tarjan ‘77, Miller ‘77, group isomorphism takes time Lipton-Synder-Zalcstein ‘78. n O (log n ) . Proof. Homomorphisms f : G = � X � → H are set f : X → H . So | hom( G, H ) | ≤ | H | d ( X ) . 2 If Q = � X � , then ∀ x ∈ X , Q = � X − { x }� . So d ( G ) cannot be ordinal.

10 Isomorphism for unbiased order is usually easy! DRAFT Theorem. H¨ older 1895. Theorem (W.) ∀ ǫ > 0 , ∃ d Groups of square free order are such that group isomorphism can be decided in time O ( n d ) Z a ⋊ Z b , ( a, b ) = 1. for a set of finite cardinals of density (1 − ε ). Theorem. Slattery ‘04, Groups of order n = p 1 · · · p s (E.g. O ( n 8 ) covers 99 . 6% of all have O ((log n ) c )-time isomor- group orders.) phism tests. Proof. Guralnick ‘89, Luc- chini 2000, show if n = p e 1 1 · · · p e s s , p i prime, then Theorem. Dietrich-Eick 2005 d ( G ) ≤ µ ( N ) := max { e i } . Same for cube-free. The number of integers n with µ ( n ) < d tends to 1 /ζ ( d ). � To be fair, you had to factor N .

11 Besche-Eick-O’Brien 2000. DRAFT 1e+12 N = 2 10 1e+10 N = 2 9 · 3 1e+08 N = 2 9 log f ( N ) 1e+06 10000 100 1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 N A log-scale plot of the number f ( N ) of the groups of order N .

12 (Probably) most finite groups order 2 k , 2 k 3 , 3 k .... DRAFT Conjecture. Erd˝ os Fact. The number of graphs Up to isomorphism most on n vertices is 2 Θ( n 2 ) . groups of size ≤ n have order 2 m . Theorem. Higman 60; Sims 65 Fact. The number of semi- The number f ( p m ) of groups of groups of order n vertices is order p m is 2 Θ( n 2 log n ) . p 2 m 3 / 27+Ω( m 2 ) ∩ O ( m 3 − ǫ ) Groups do not grow like com- for a some ǫ > 0. binatorics. The rare prime Theorem. Pyber 93 The power sized sets are by far the number f ( n ) of groups order at most complex. n satisfies f ( n ) ≤ n 2 µ ( n ) 2 / 27+ Dµ ( n ) 2 − ǫ .

13 What grows like groups? DRAFT Theorem. Kruse-Price-70 Theorem. Poonen-08 The number of finite rings of or- The number of commutative der p m is rings of order p m is p 2 m 3 / 27+Ω( m 2 ) ∩ O ( m 3 − ǫ ) p 4 m 3 / 27+Ω( m 2 ) ∩ O ( m 3 − ǫ ) . Why so similar to groups? Theorem. Neretin-87 Hint. The dimension of the variety of Groups have a second product algebras is [ x, y ] = x − 1 x y = x − 1 y − 1 xy 2 27 n 3 + D 1 m 3 − ǫ 1 and it nearly distributes: [ xy, z ] = [ x, z ] y [ y, z ] . for commutative or Lie, 4 27 m 3 + D 2 m 3 − ǫ 2 for associative.

14 Step one: separate nilpotent from reductive DRAFT ֒ → − → Step two: Break nilpotent into abelian sections �

15 Where is the complexity in “triangular matrices”? DRAFT B. Matrix type groups A. Nonassociative products � s u w � � s ′ u ′ w ′ need 3-dimensional array of pa- � 0 s ′ v ′ = 0 s v rameters. Entropy of Θ( m 3 ). 0 0 s 0 0 s ′ � ss ′ us ′ + su ′ ws ′ + u ∗ v ′ + sw ′ � ss ′ vs ′ + sv ′ 0 ss ′ 0 0 need only ∗ : U × V ֌ W . d ( U ) d ( V ) d ( W ) n d ( U ) d ( V ) d ( W ) ≤ m 3 / 27 n n

16 C. Cut to diagonal embedding D. Add symmetry DRAFT �� s u w � � : u ∈ U, w ∈ W 0 s ± uθ 0 0 s need ± θ -Hermitian     s u w  : u ∈ U,   ∗ : U × U ֌ W. 0 s ± uθ  w ∈ W d ( U ) 0 0 s   d ( W ) now use ∗ : U × U ֌ W . d ( U ) 1 2 d ( U ) 2 ( n − d ( U )) ≤ 2 n 3 / 27. d ( W ) d ( U ) 2 ( m − d ( U )) ≤ 4 m 3 / 27.

17 DRAFT Moral: Isomorphism of your groups might be easy. But most groups are made the same way as rings and algebras. It is all about bilinear maps ∗ : U × V ֌ W and the Hermitian ones. Open problem: Decide if two bimaps are isotopic/pseudo- isometric.

18 (Brooksbank-W.) The adjoint-tensor attack DRAFT Theorem. W.-Lewis. Proof. Fix ∗ : U × V ֌ W . Quotients of Heisenberg groups M ∗ = { ( f, g ) : uf ∗ v = u ∗ gv } . over fields have O ((log n ) 6 )- time isomorphism tests, this despite having no known group Fact. ∗ factors through ⊗ M ∗ theoretic differences. and this is the smallest possible tensor product for ∗ . Theorem. Brooksbank-W. Central products of quotients Aut( ∗ ) is a stabilizer in of Heisenberg groups over Aut( ⊗ M ∗ ) and Aut( ⊗ M ∗ ) is cyclic rings have O ((log n ) 6 )- the normalizer of M ∗ . time isomorphism tests. If the rings M ∗ are In both cases these handle semisimple then computed ef- p cm 2 many groups. ficiently. �

19 (W.) Triality attack. DRAFT T ( L ∗ ⊘ ) T ( ⊘ ) T ( ⊗ ) T ( ∗ ) T ( ⊘ ) T ( ⊗ M ∗ ) T ( ⊘ R ∗ )

� 20 Theorem (Why the triality attack works). W. DRAFT There are exact sequences 1 →L × ∗ → Aut( ∗ ) → Aut( V ∗ ) 1 →M × ∗ → Aut( ∗ ) → Aut( W ∗ ) 1 →R × ∗ → Aut( ∗ ) → Aut( U ∗ ) 1 → Aut C ∗ ( ∗ ) → Aut( ∗ ) → Out( C ∗ ) and � LMR × � Z ( LMR ∗ ) × 1 ∗ × Aut LMR ( ∗ ) � Aut C ∗ ( ∗ ) Out C ∗ ( LMR ∗ ) . If e 2 = e ∈ LMR ∗ such that LMR ∗ = LMR ∗ e LMR ∗ then Aut LMR ( ∗ : U × V ֌ W ) ∼ = Aut( eUe × eV e ֌ eWe ) .

21 (Maglione-W.) The filter attack DRAFT Theorem. W. Theorem. Maglione Every group (and every There is a polynomial-time al- ring/algebra) can be given a gorithm to compute this filter. filter where the homogeneous products Survey. Maglione-W. Of the 11 million groups of or- ∗ : H i × H j ֌ H i + j der ≤ 1000, over 81% admit a proper decomposition by these each have LMR ∗ semisimple. filters. Theorem. W. A positive logarithmic propor- tion of all finite groups admit proper refinements.

22 76 , . . . , a p G = � a 1 , . . . , a 76 : [ a 1 , a 2 ] = a ∗ 3 · · · a ∗ 1 = a ∗ 2 · · · a ∗ 76 , . . . � DRAFT Naive lower central series. Rediscovered “matrix” configu- ration Refinement breaks into smaller and structured parts.