Large element orders and the characteristic of finite simple - - PowerPoint PPT Presentation

Large element orders and the characteristic of finite simple symplectic and orthogonal groups

Daniel Lytkin

Novosibirsk State University

Groups St Andrews 3rd – 11th August 2013

Dan Lytkin (Novosibirsk) 1 / 10

Matrix group recognition

Matrix group recognition

Let G = X ≤ GL(n, q) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group.

Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition

Matrix group recognition

Let G = X ≤ GL(n, q) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process

Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition

Matrix group recognition

Let G = X ≤ GL(n, q) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process

• 1. find the characteristic of G
• ch(G)
• Dan Lytkin

(Novosibirsk) 2 / 10

Matrix group recognition

Matrix group recognition

Let G = X ≤ GL(n, q) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process

• 1. find the characteristic of G
• ch(G)
• 2. determine the stardard name of G

Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition

Matrix group recognition

Let G = X ≤ GL(n, q) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process

• 1. find the characteristic of G
• ch(G)
• 2. determine the stardard name of G
• 3. produce an explicit authomorphism with a known simple group

Dan Lytkin (Novosibirsk) 2 / 10

Matrix group recognition

Matrix group recognition

Let G = X ≤ GL(n, q) be a matrix group specified by a set X of generators, known to be isomorphic to a Lie-type simple group. Recognition process

• 1. find the characteristic of G
• ch(G)
• 2. determine the stardard name of G
• 3. produce an explicit authomorphism with a known simple group

In the 2009 paper by Kantor and Seress a Monte Carlo algorithm is described for finding the characteristic of Lie-type simple groups. It involves examining the orders of a random selection of group elements.

Dan Lytkin (Novosibirsk) 2 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G.

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified)

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0.

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅;

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

• 1. g := random element of G, place |g| in L; repeat up to N times;

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

• 1. g := random element of G, place |g| in L; repeat up to N times;

random sample of size N contains elements of orders m1(G), m2(G), m3(G) with probability at least 1 − ε

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

• 1. g := random element of G, place |g| in L; repeat up to N times;

random sample of size N contains elements of orders m1(G), m2(G), m3(G) with probability at least 1 − ε There is N that depends only on n and ε!

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Denote by m1(G) > m2(G) > . . . the largest element orders of a group G. Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

• 1. g := random element of G, place |g| in L; repeat up to N times;

random sample of size N contains elements of orders m1(G), m2(G), m3(G) with probability at least 1 − ε There is N that depends only on n and ε!

• 2. m′

1, m′ 2, m′ 3 := three largest numbers in L;

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

• 1. g := random element of G, place |g| in L; repeat up to N times;
• 2. m′

1, m′ 2, m′ 3 := three largest numbers in L;

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

• 1. g := random element of G, place |g| in L; repeat up to N times;
• 2. m′

1, m′ 2, m′ 3 := three largest numbers in L;

• 3. Use F to determine H with m′

1 = m1(H) and m′ 2 = m2(H). Use m′ 3 to

distinguish *ambiguous cases*;

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

• 1. g := random element of G, place |g| in L; repeat up to N times;
• 2. m′

1, m′ 2, m′ 3 := three largest numbers in L;

• 3. Use F to determine H with m′

1 = m1(H) and m′ 2 = m2(H). Use m′ 3 to

distinguish *ambiguous cases*;

• 4. Return ch(H) as output.

Dan Lytkin (Novosibirsk) 3 / 10

Finding the characteristic

Theorem (Kantor W. M., Seress A.) Let G and H be simple groups of Lie type, different from Sp2n(2k) and Ω±

2n(2k). If mi(G) = mi(H) for i = 1, 2, then one of the following holds:

• 1. ch(G) = ch(H).
• 2. *few ambiguous cases*

Algorithm (simplified) Input: G = X ≤ GL(n, q) and an error bound ε > 0. Output: ch(G).

• 0. L := ∅; F := set of formulae for triples (H, m1(H), m2(H)) for all

Lie-type simple groups H;

• 1. g := random element of G, place |g| in L; repeat up to N times;
• 2. m′

1, m′ 2, m′ 3 := three largest numbers in L;

• 3. Use F to determine H with m′

1 = m1(H) and m′ 2 = m2(H). Use m′ 3 to

distinguish *ambiguous cases*;

• 4. Return ch(H) as output.

We don’t know how to compute m1(G) and m2(G) if G is Sp2n(2k) or Ω±

2n(2k).

Dan Lytkin (Novosibirsk) 3 / 10

Element orders of symplectic groups

Symplectic groups

Denote by S(n, q) the set of all numbers [qn1 ± 1, qn2 ± 1, . . . , qns ± 1] for all integer partitions n1 + n2 + · · · + ns = n.

Dan Lytkin (Novosibirsk) 4 / 10

Element orders of symplectic groups

Symplectic groups

Denote by S(n, q) the set of all numbers [qn1 ± 1, qn2 ± 1, . . . , qns ± 1] for all integer partitions n1 + n2 + · · · + ns = n. If q is a power of 2 then the set all element orders of Sp2n(q) consists of all divisors of the following numbers:

• S(n, q)
• 2 · S(n − 1, q)
• 2k · S(n − 1 − 2k−2, q) for k 2
• 2k if 2k−2 + 1 = n for k 2

Dan Lytkin (Novosibirsk) 4 / 10

Element orders of symplectic groups

Symplectic groups

Denote by S(n, q) the set of all numbers [qn1 ± 1, qn2 ± 1, . . . , qns ± 1] for all integer partitions n1 + n2 + · · · + ns = n. If q is a power of 2 then the set all element orders of Sp2n(q) consists of all divisors of the following numbers:

• S(n, q)
• 2 · S(n − 1, q)
• 2k · S(n − 1 − 2k−2, q) for k 2
• 2k if 2k−2 + 1 = n for k 2

Buturlakin A. A. Spectra of finite symplectic and orthogonal groups // Siberian Advances in Mathematics, 2011, V 21, N 3

Dan Lytkin (Novosibirsk) 4 / 10

Element orders of symplectic groups

Proposition Let G = Sp2n(q) where n odd, q = 2k > 2 and 2a1 + 2a2 + · · · + 2at + 2r + 2p is the binary expansion of n in ascending order. Then m1(G) = [q2a1 + 1, . . . , q2at + 1, q2r + 1, q2p + 1], m2(G) = [q2a1 + 1, . . . , q2at + 1, q3·2r + 1, q2r+1 + 1, . . . , q2p−1 + 1]

Dan Lytkin (Novosibirsk) 5 / 10

Element orders of symplectic groups

Proposition Let G = Sp2n(q) where n odd, q = 2k > 2 and 2a1 + 2a2 + · · · + 2at + 2r + 2p is the binary expansion of n in ascending order. Then m1(G) = [q2a1 + 1, . . . , q2at + 1, q2r + 1, q2p + 1], m2(G) = [q2a1 + 1, . . . , q2at + 1, q3·2r + 1, q2r+1 + 1, . . . , q2p−1 + 1] Proposition Let G = Sp2n(q) where n even, q = 2k > 2. Then

• 1. If n = 2 then m1(G) = q2 + 1, m2(G) = q2 − 1
• 2. If n = 4 then m1(G) = [q + 1, q3 − 1], m2(G) = q4 + 1
• 3. If n > 4 then

m1(G) = [q + 1, q2 + 1, . . . , q2u + 1, qn−2u+1+1 − 1], m2(G) =

• [q + 1, q2 + 1, . . . , q2u+1 + 1, qn−2u+2+1 − 1],

if 5 · 2u n [q + 1, q2 + 1, . . . , q2u−1 + 1, qn−2u+1 − 1],

• therwise

where u is the largest integer such that 2u n/3.

Dan Lytkin (Novosibirsk) 5 / 10

Element orders of symplectic groups

Table : Two largest element orders of groups G = Sp2n(2) Conditions m1(G) m2(G) n = 3 15 12 n = 4 30 24 n = 5 60 51 n = 6 120 105 n = 2l − 1 7 2n+1 − 1 2(2

n+1 2

− 1)(2

n−1 2

− 1) n = 2l 8 2(2n − 1) (2

n 2 − 1)(2 n 2 +1 − 1)

n = 2l + 1 9 4(2n−1 − 1) 2(2

n+1 2

− 1)(2

n−1 2

− 1) n = 2l + 2 10 8(2n−2 − 1) (2

n 2 −1 − 1)(2 n 2 +2 − 1)

n = 3 · 2l − 1 11 (2

2n+2 3

+ 1)(2

n+1 3

− 1) 2(2

n+1 3

− 1)(2

2n−1 3

− 1) n = 3 · 2l 12 (2

2n 3 − 1)(2 n 3 +1 − 1)

2(2

n 3 − 1)(2 2n 3 + 1)

n = 3 · 2l + 1 13 2(2

2n−2 3

− 1)(2

n+2 3

− 1) 4(2

n−1 3

− 1)(2

2n−2 3

+ 1) n = 3 · 2l + 2 14 (2

2n−4 3

− 1)(2

n+7 3

− 1) 4(2

2n−4 3

− 1)(2

n+1 3

− 1) Other odd n 2(22p+1 − 1)(2n−2p+1 − 1) 8(22p+1 − 1)(2n−2p+1−2 − 1) Other even n (22p+1 − 1)(2n+1−2p+1 − 1) 4(22p+1 − 1)(2n−1−2p+1 − 1)

Dan Lytkin (Novosibirsk) 6 / 10

Element orders of symplectic groups

Theorem Let G = Sp2n(q) where q = 2k and H is a group of Lie type such that mi(G) = mi(H) for i = 1, 2. Then ch(H) = 2.

Dan Lytkin (Novosibirsk) 7 / 10

Element orders of orthogonal groups

Orthogonal groups

Dan Lytkin (Novosibirsk) 8 / 10

Element orders of orthogonal groups

Orthogonal groups

Denote by Sε(n, q) with ε ∈ {+, −} the set of all numbers [qn1 + 1, qn2 + 1, . . . , qnl + 1, qnl+1 − 1, qnl+2 − 1, . . . , qnl − 1] for all integer partitions n1 + n2 + · · · + ns = n and l even if ε = + and odd

• therwise.

Dan Lytkin (Novosibirsk) 8 / 10

Element orders of orthogonal groups

Orthogonal groups

Denote by Sε(n, q) with ε ∈ {+, −} the set of all numbers [qn1 + 1, qn2 + 1, . . . , qnl + 1, qnl+1 − 1, qnl+2 − 1, . . . , qnl − 1] for all integer partitions n1 + n2 + · · · + ns = n and l even if ε = + and odd

• therwise.

If q is a power of 2, ε ∈ {+, −} then the set all element orders of Ωε

2n(q)

consists of all divisors of the following numbers:

• Sε(n, q)
• 2 · S(n − 2, q)
• 2[q ± 1, Sε(n − 2, q)]
• 4[q − 1, qn1 + 1, qn2 + 1, . . . , qns + 1] for integer partitions

n1 + n2 + · · · + ns = n − 3 with s even if ε = + and odd otherwise

• 4[q + 1, S−ε(n − 3, q)]
• 2k · S(n − 2 − 2k−2, q) for k 2
• 2k if 2k−2 + 2 = n for k > 2

Dan Lytkin (Novosibirsk) 8 / 10

Element orders of orthogonal groups

Introduce the notation: µk(n, q) = (q + 1) . . . (q2k + 1)(qn−2k+1−1 − 1) ηk(n, q) = (q + 1) . . . (q2k−2 + 1)(q2k + 1)(qn−3·2k−1+1 − 1) Proposition Let n 10 even, q = 2k > 2, α integer, such that 2α n/3 < 2α+1 and ε = + for even α and ε = − otherwise. Then the following table is correct: Conditions m1 (Ωε

2n(q)) m2 (Ωε 2n(q)) m1

• Ω−ε

2n (q)

• m2
• Ω−ε

2n (q)

• n < 7 · 2α−1

µα−1(n, q) ηα(n, q) µα(n, q) µα−2(n, q) 7 · 2α−1 n < 9 · 2α−1 µα−1(n, q) ηα(n, q) µα(n, q) ηα+1(n, q) 9 · 2α−1 n < 5 · 2α µα−1(n, q) µα+1(n, q) µα(n, q) ηα+1(n, q) n 5 · 2α µα+1(n, q) µα−1(n, q) µα(n, q) ηα+1(n, q)

Dan Lytkin (Novosibirsk) 9 / 10

Further steps

What is left to do

Dan Lytkin (Novosibirsk) 10 / 10

Further steps

What is left to do

• Maximal element orders for remaining orthogonal groups

Dan Lytkin (Novosibirsk) 10 / 10

Further steps

What is left to do

• Maximal element orders for remaining orthogonal groups
• ch-property for orthogonal groups

Dan Lytkin (Novosibirsk) 10 / 10

Further steps

What is left to do

• Maximal element orders for remaining orthogonal groups
• ch-property for orthogonal groups
• Proportion of maximal-order elements

Dan Lytkin (Novosibirsk) 10 / 10