on (2,3)-generated groups - nsc.ru · 2013. 7. 30. · maxim vsemirnov steklov institute of...

Definitions and motivations(2,3)-generated finite (simple) groups

(2,3)-generated classical groups over Z

On (2,3)-generated groups

Maxim Vsemirnov

Steklov Institute of Mathematics at St. Petersburg

Group Theory Conference in honor of V. D. MazurovNovosibirsk, July 20, 2013

M. Vsemirnov On (2,3)-generated groups



Outline

1 Definitions and motivations

2 (2,3)-generated finite (simple) groups

3 (2,3)-generated classical groups over Z




(2,3)-generated groups

DefinitionA (2,3)-generated group is a group generated by an involutionand an element of order 3.

DefinitionAn (m,n)-generated group is a group generated by twoelements of order m and n, respectively.

Why is the (2,3)-generation problem interesting?




Why do we look at (2, 3)?

The modular group PSL2(Z) is isomorphic to the free product oftwo cyclic groups, C2 and C3.

PSL2(Z) =〈(

0 −11 0

),

(0 −11 −1

)〉' C2 ∗ C3.

Thus, apart from {1}, C2, and C3, all quotients of PSL2(Z) areexactly the (2,3)-generated groups.




Comparison with PSLn(Z), n ≥ 3.

RemarkThe normal subgroup structure of PSL2(Z) differs dramaticallyfrom the normal subgroup structure of PSLn(Z), n ≥ 3.

Namely, for n ≥ 3, any subgroup of finite index in PSLn(Z) is aso-called congruence subgroup, i.e., contains the kernel ofPSLn(Z) 7→ PSLn(Z/mZ) for some m.

In contrast, PSL2(Z) contains many noncongruence normalsubgroups (even of finite index).




Outline







Two different approaches

There are two different groups of methods in this area:

constructive, i.e., when the corresponding generators aregiven explicitly;non-constructive, e.g., probabilistic, when only existencetheorems are known (usually require a good knowledge ofthe characters and maximal subgroups).

Constructive methods can be also applied to infinite groups.




Known results

1 For any m, PSL2(Z/mZ) is (2,3)-generated (trivial)2 An, n ≥ 4, are (2,3)-generated except A6, A7 , and A8

(Miller, 1901).3 Sporadic groups are (2,3)-generated except M11, M22, M23,

and McL (Woldar, 1989)4 2B2(22k+1) are not (2,3)-generated (trivial, since they do

not contain elements of order 3)5 Other exceptional Lie groups are (2,3)-generated (Malle,

1990, 1995, Malle and Lübeck, 1999)




Classical groups and the (2,3)-generation

Negative results for certain small groups:PSL2(9) ' Sp4(2)′ ' A6, PSL4(2) ' A8, PSL3(4), PSU3(9).

For n large enough, SLn(q) are (2,3)-generated;Tamburini, J. Wilson (1994–1995), n ≥ 14;Di Martino, Vavilov (1994–1996) for n ≥ 5, q 6= 3, q 6= 2k .

PSp4(pk ) are (2,3)-generated if p 6= 2,3 (Di Martino and

Cazzola, 1993).

PSp4(pk ) are not (2,3)-generated if p = 2,3 (Liebeck and

Shalev, 1996).

Almost all classical groups are (2,3)-generated (Liebeck andShalev, 1996).




Probabilistic methods

Theorem (Liebeck, Shalev, 1996)Let G run through some infinite set of finite classical groups,G 6= PSp4(pk ). Then

lim|G|→∞

Prob(x2 = y3 = 1 and G = 〈x , y〉) = 1.

Moreover, the result remains trueif we fix the field and let the rank tend to infinity;if we fix the type and let the size of the field tend to infinity.




New examples of non (2,3)-generated groups

Theorem (V., 2011)

PSU5(4) is not (2,3)-generated.

|PSU5(4)| = 13,685,760 = 210 · 35 · 5 · 11

A sketch of the proof.1. dim ker(x − 1) = 3 and y ∼ diag(1, ω, ω, ω−1, ω−1),ω2 + ω + 1 = 0.

x =

0 1 0 0 c1 0 0 0 −c0 0 0 1 d0 0 1 0 −d0 0 0 0 1

, y =

1 0 a 0 b0 0 −1 0 00 1 −1 0 00 0 0 0 −10 0 0 1 −1

,for some a, b, c, d .




A sketch of the proof (cont.)

2. If det

3 + a ac + bd − b a + b−1 b + cd + c c − a− 1−1 1− c + d2 + d 1 + d

= 0, then〈x , y〉 has a 1-dimensional invariant space.

3. If 〈x , y〉 preserves a hermitian form thena = −d − dσ − 1,b = −c + cσ + d + dσ + d2 + ddσ − 1.

4. There are 16 pairs of parameters (c,d).For four of them, the group is defined over F2.For ten of them, det(· · · ) = 0.For the remaining two, setting z = yx we have

z11 = x2 = (zx)3 = (z4xz6x)2 = 1,

a well-known presentation of PSL2(11).M. Vsemirnov On (2,3)-generated groups



New examples of non (2,3)-generated groups

Theorem (Pellegrini, Tamburini Bellani, V., 2012)

PSU4(9) is not (2,3)-generated.

Theorem (V., 2012)

Ω+8 (2), PΩ+8 (3) are not (2,3)-generated.

|PSU4(9)| = 3,265,920 = 27 · 36 · 5 · 7|Ω+8 (2)| = 174,182,400 = 2

12 · 35 · 52 · 7|PΩ+8 (3)| = 4,952,179,814,400 = 2

12 · 312 · 52 · 7 · 13




Non (2,3)-generated finite simple groups

1 PSp4(2k )

2 PSp4(3k ), in particular PSp4(3) ' PSU4(4)

3 2B2(22k+1)4 A6 ' PSL2(9) ' Sp4(2)′, A7, A8 ' PSL4(2)5 PSL3(4), PSU3(9) ' G2(2)′6 M11, M22, M23, McL7 PSU5(4)8 PSU4(9)9 Ω+8 (2), PΩ

+8 (3)

10 ?

I strongly believe that the list is complete.




Outline







The main Theorem

TheoremThe groups SLn(Z) and GLn(Z) are (2,3)-generated preciselywhen n ≥ 5




An overview of results

SL2(Z) is not (2,3)-generated as it contains no non-centralinvolution.SL4(Z) and GL4(Z) are not (2,3)-generated asSL4(2) = GL4(2) ' A8 is not (Miller, 1901)SL3(Z) and GL3(Z) are not (2,3)-generated (Nuzhin, 2001,Tamburini, Zucca, 2001).SLn(Z) and GLn(Z) are (2,3)-generated for n ≥ 14(Tamburini, et al. 1994–1995, 2009)For SL5(Z), GL5(Z) and SL6(Z) there are at most finitelymany conjugacy classes of (2,3)-generators (Luzgarev,Pevzner, 2003, Vsemirnov, 2006).The groups SLn(Z) and GLn(Z), n = 5, . . . ,13 are(2,3)-generated (Vsemirnov, 2007–2009).




An idea of the proof

Two difficult problems:

to guess the shape of (2,3)-generators;to show that they actually generate SLn(Z).

The main idea: show that 〈x , y〉 contains some generating setof SLn(Z).

For instance, one can show that 〈x , y〉 contains elementarytransvections tij(α) = I + αeij , i 6= j .




One rather complicated example

x =

−3 0 4 0 4−1 −1 4 0 0−1 0 2 0 1

2 0 −2 −1 −6−1 0 1 0 2

, y =

2 −2 0 1 13 −2 −1 1 −22 −1 −1 1 0−1 2 −2 −1 −3

0 0 0 0 1

.

h1 = (yx)3(y2x)3yxy2x ,h2 = h−41 = t52(2),

h3 = yxy2xyxy2,h4 = yxyxyxy2xyxyxy2,. . .

h51 = h47h48h49h50h62h−120 h

−1321 h

−1529 h

1512h

−836 = t53(1)




Two special cases

Let M = Matn(Q). If 〈x , y〉 is absolutely irreducible then

dim CM(x) + dim CM(y) + dim CM(xy) ≤ n2 + 2.

Further analysis depends on whether

dim CM(x) + dim CM(y) + dim CM(xy) < n2 + 2

ordim CM(x) + dim CM(y) + dim CM(xy) = n2 + 2.

In the latter case it is possible to classify all (2,3)-generatingpairs of SLn(Z) up to conjugation. This happens precisely whenn = 5 and n = 6.




SL5(Z)

Theorem (V., 2007)

Any (2,3)-generating pair of SL5(Z) is conjugate in GL5(Z) toone of the pairs −X , Y , and any (2,3)-generating pair ofGL5(Z) is conjugate to one of the pairs X , Y , where

X =

−1 0 0 0 0

0 −1 0 0 00 0 −1 0 01 0 0 1 00 0 1 0 1

, Y =

0 1 0 0 a1−1 −1 0 0 a2

0 0 0 1 a30 0 −1 −1 a40 0 0 0 1

,

and (a1,a2,a3,a4) is one of the sets

(1,−1,−2,−2), (0,−1,−2,−2),(−1,1,−2,−2), (0,1,−2,−2),

(1,−1,1,−3), (0,−1,0,−1).M. Vsemirnov On (2,3)-generated groups



SL6(Z)

Theorem (V., 2012)

Any (2,3)-generating pair of SL6(Z) is conjugate in GL6(Z) toone of the pairs ±X , Y , where

X =

0 I2 BI2 0 −B0 0 I2

, Y = I2 0 A0 0 −I2

0 I2 −I2

,A =

(a1 a2a3 a4

), B =

(b1 b2b3 b4

),

and (b1,b2,b3,b4,a1,a2,a3,a4) is either(0,2,−2,−3,3,1,−1,1) or (1,−3,3,−4,1,1,−1,3).




A version of the ping-pong lemma

Lemma (V., 2007)

Let x , y ∈ GLn(Z), n > 3, x2 = y3 = I. Assume that for someW ⊆ Rn and w ∈ Rn \W, we have(i) xyW ⊆W, xy2W ⊆W ;(ii) xy · w ∈ W, xy2 · w ∈ W.Then 〈x , y〉 ' PSL2(Z). In particular, 〈x , y〉 6= GLn(Z),〈x , y〉 6= SLn(Z).




Symplectic case

Theorem (Vasiliev, Vsemirnov, 2008–2011)

Sp2(Z), Sp4(Z), and Sp6(Z) are not (2,3)-generated.Sp8(Z), Sp10(Z) are (2,3)-generated.Sp2n(Z) are (2,3)-generated for n ≥ 25.

Cases 2n = 12,14, . . . ,48 remain open. We expect the positiveanswer.


Definitions and motivations(2,3)-generated finite (simple) groups(2,3)-generated classical groups over Z

on (2,3)-generated groups - nsc.ru · 2013. 7. 30. · maxim vsemirnov steklov institute of...

Documents