OF A GENERATING SET

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M AXIMAL SUBGROUPS OF FINITE GROUPS AVOIDING THE ELEMENTS

OF A GENERATING SET

Andrea Lucchini

Università di Padova, Italy

Groups and Topological Groups 2017 Budapest, February 3rd, 4th

Joint work withPablo Spiga

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A N EASY THEOREM

THEOREM

Let G be a finitely generated group and let d = d (G) be the smallest cardinality of a generating set of G. If G = hg₁, . . . ,g_di, then there exists a maximal subgroup M of G such that M ∩ {g₁, . . . ,g_d} = ∅.

PROOF.

If G is cyclic the statement is clear. When d > 1, consider H = hg1g2,g2g3, . . . ,gd −1gdi.

d (H) ≤ d − 1 < d (G) ⇒ H 6= G. Let M be a maximal subgroup of G containing H.

g_i ∈ M and i 6= d ⇒ g_i+1=g_i⁻¹(g_ig_i+1) ∈M. g_i ∈ M and i 6= 1 ⇒ g_i−1 = (g_i−1g_i)g_i⁻¹∈ M. Thus M ∩ {g₁, . . . ,g_d} 6= ∅ implies G = hg₁, . . . ,g_di ≤ M, a contradiction.

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A N EASY THEOREM

THEOREM

Let G be a finitely generated group and let d = d (G) be the smallest cardinality of a generating set of G. If G = hg₁, . . . ,g_di, then there exists a maximal subgroup M of G such that M ∩ {g₁, . . . ,g_d} = ∅.

PROOF.

If G is cyclic the statement is clear. When d > 1, consider H = hg1g2,g2g3, . . . ,gd −1gdi.

d (H) ≤ d − 1 < d (G) ⇒ H 6= G.

Let M be a maximal subgroup of G containing H.

g_i ∈ M and i 6= d ⇒ g_i+1=g_i⁻¹(g_ig_i+1) ∈M.

g_i ∈ M and i 6= 1 ⇒ g_i−1 = (g_i−1g_i)g_i⁻¹∈ M.

Thus M ∩ {g₁, . . . ,g_d} 6= ∅ implies G = hg₁, . . . ,g_di ≤ M, a contradiction.

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This result does not remain true if we drop the assumption d = d (G).

For example, let G = F^d₂be the additive group of a vector space of dimension d ≥ 2 over the field F2with 2 elements and let

g1= (1, 0, . . . , 0), . . . , gd = (0, . . . , 0, 1), gd +1= (1, 1, 0, . . . , 0).

Let M = {(x1, . . . ,xd) ∈ F^d₂ | a₁x1+ · · · +adxd =0} be a maximal subgroup of G. If i ≤ d , then gi ∈ M only when a_i =0. Therefore

M = {(x1, . . . ,xd) ∈ F^d2 | x₁+ · · · +xd =0}

is the unique maximal subgroup of G with gi ∈ M for every i ≤ d ./ However g_{d +1}∈ M. Hence every maximal subgroup of G contains at least one of the d + 1 elements g₁, . . . ,g_{d +1}.

Moreover,it is not sufficient to assume that {g1, . . . ,gd} is a minimal generating set of G(i.e. no proper subset of {g1, . . . ,gd} generates G): for example, if G = hx i is a cyclic group of order 6, then {x²,x³} is a minimal generating set of G, and hx²i and hx³i are the unique maximal subgroups of G.

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This result does not remain true if we drop the assumption d = d (G).

For example, let G = F^d₂be the additive group of a vector space of dimension d ≥ 2 over the field F2with 2 elements and let

g1= (1, 0, . . . , 0), . . . , gd = (0, . . . , 0, 1), gd +1= (1, 1, 0, . . . , 0).

Let M = {(x1, . . . ,xd) ∈ F^d₂ | a₁x1+ · · · +adxd =0} be a maximal subgroup of G. If i ≤ d , then gi ∈ M only when a_i =0. Therefore

M = {(x1, . . . ,xd) ∈ F^d2 | x₁+ · · · +xd =0}

is the unique maximal subgroup of G with gi ∈ M for every i ≤ d ./ However g_{d +1}∈ M. Hence every maximal subgroup of G contains at least one of the d + 1 elements g₁, . . . ,g_{d +1}.

Moreover,it is not sufficient to assume that {g1, . . . ,gd} is a minimal generating set of G(i.e. no proper subset of {g1, . . . ,gd} generates G): for example, if G = hx i is a cyclic group of order 6, then {x²,x³} is a minimal generating set of G, and hx²i and hx³i are the unique maximal subgroups of G.

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THEOREM

Let G be a finitely generated group and let d = d (G) be the smallest cardinality of a generating set of G. If G = hg1, . . . ,gdi, then there exists a maximal subgroup M of G such that M ∩ {g₁, . . . ,g_d} = ∅.

The proof of this theorem is extremely easy, but it does not give any insight on the freedom that we have in the choice of the maximal subgroup M.

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T HE SOLUBLE CASE

NOTATIONS

M a maximal subgroup of a finite soluble group G, Y_M =T

g∈GM^gthe normal core of M in G,

X_M/Y_M the socle of the primitive permutation group G/Y_M.

XM

YM is a chief factor of G and_Y^M

M is a complement of ^X_Y^M

M in_Y^G

M. V:= a set of representatives of the irreducible G-modules that are G-isomorphic to some chief factor of G having a complement.

for every V ∈ V, letM_V be the set of maximal subgroups M of G with X_M/Y_M∼=_GV .

QUESTION

For which V ∈ V, does there exist M ∈ MV with M ∩ {g1, . . . ,gd} = ∅?

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It is useful to recall some results by Gaschütz. Given V ∈ V, let RG(V )= \

M∈MV

M.

R_G(V ) is the smallest normal subgroup of G contained in C_G(V ) with C_G(V )/R_G(V ) being G-isomorphic to a direct product of copies of V and having a complement in G/RG(V ).

The factor group C_G(V )/R_G(V ) is called theV -crownof G. The non-negative integerδ_G(V )defined by

C_G(V ) R_G(V )

∼=_GV^δ^G^{(V )}

is called theV -rankof G and it equals the number of complemented factors in any chief series of G that are G-isomorphic to V .

G/RG(V ) ∼= V^δ^G^{(V )}oG/CG(V )where G/CG(V ) acts diagonally on V^δ^G^{(V )}.

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M∈MV

M.

R_G(V ) is the smallest normal subgroup of G contained in C_G(V ) with C_G(V )/R_G(V ) being G-isomorphic to a direct product of copies of V and having a complement in G/R_G(V ).

The factor group C_G(V )/R_G(V ) is called theV -crownof G. The non-negative integerδ_G(V )defined by

C_G(V ) R_G(V )

∼=_GV^δ^G^{(V )}

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M∈MV

M.

The factor group C_G(V )/R_G(V ) is called theV -crownof G.

The non-negative integerδ_G(V )defined by C_G(V )

R_G(V )

∼=_GV^δ^G^{(V )}

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M∈MV

M.

R_G(V )

∼=_GV^δ^G^{(V )}

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M∈MV

M.

R_G(V )

∼=_GV^δ^G^{(V )}

G/R_G(V ) ∼= V^δ^G^{(V )}oG/CG(V )where G/C_G(V ) acts diagonally on V^δ^G^{(V )}.

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Let G = hg1, . . . ,gdi. We want to check whether there exists M ∈ M_V with M ∩ {g1, . . . ,gd} = ∅. We write G = G/R_G(V ) and, for every g ∈ G, we denote by g the element gRG(V ) of ¯G. Letδ= δ_G(V ).

CASE1: V IS A TRIVIALG-MODULE.

G = CG(V ),G ∼= F^δpand elements in MV are in one-to-one correspondence with the hyperplanes of F^δp.

For every i, we identify ¯g_i with the vector (x_i1, . . . ,x_iδ)of F^δp. A maximal subgroup M of ¯G is determined by a linear equation a₁x₁+ · · · +aδxδ=0 and ¯g_i ∈ M if and only ifPδ

j=1a_jx_ij =0.

The linear map φ : F^δp→ F^dp defined by setting

φ(a1, . . . ,aδ) =





δ

X

j=1

ajx1j, . . . ,

δ

X

j=1

ajxdj





is injective.

The existence of M ∈ M_V with M ∩ {g₁, . . . ,g_d} = ∅ is equivalent to the existence of (b₁, . . . ,b_d) ∈ φ(F^δp)with b_i 6= 0 for every i.

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CASE2: V IS A NON-TRIVIALG-MODULE.

H :=G/C_G(V ),F:=End_G(V ). We haveG ∼= V^δo H.

We may write ¯g_i =h_iw_i with h_i ∈ H and w_i = (v_i1, . . . ,v_iδ) ∈V^δ. Ω:=V × F^δandΩ^∗:= {(w , 0, . . . , 0) ∈ Ω | w ∈ V }.

There is a one-to-one correspondence between the elements of M_V and the 1-dimensional subspaces of Ω contained in Ω \ Ω^∗: for every ω = (w , λ₁, . . . , λ_δ) ∈ Ω \ Ω^∗, we associate the maximal subgroup Mω= {h(v₁, . . . ,vδ) ∈ ¯G | w − w^h+Pδ

j=1λ_jv_j =0}.

An injective linear mapφ: Ω →V^d is defined by setting

φ(w , λ₁, . . . , λ_δ) =







w − w^h¹+

δ

X

j=1

λ_jv_1j



,. . .,



w − w^h^d+

δ

X

j=1

λ_jv_dj









The M ∈ MV with M ∩ {g1, . . . ,gd} = ∅ correspond to the ω ∈ Ω \ Ω^∗s.t. φ(ω) = (v1, . . . ,vd)has all non-zero coordinates.

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THEOREM

Let G = hg₁, . . . ,g_di be a finite soluble group with d = d (G) and let V ∈ V. Set θ_G(V ) = 1 if V is a non-trivial G-module and θ_G(V ) = 0 otherwise, FV =End_G(V ), q_V = |FV| and n_V =dim_F_V(V ). If

δ_G(V ) ≥ (d − 1 − θ_G(V ))n_V+1,

then there exists M ∈ M_V with M ∩ {g₁, . . . ,g_d} = ∅. Moreover, if there exists a unique choice for M, then one of the following occurs:

(1) V is a trivial G-module, q_V =2 and δ_G(V ) = d ; (2) V is a non-trivial G-module, d = 2, δ_G(V ) = 1 and

(qV,nV) ∈ {(3, 1), (2, 2)}.

LEMMA

Let V₁, . . . ,V_d be vector spaces of dimension n over Fq. Assume d ≥ 2 and, when q = 2, assume also n ≥ 2. Let W be a subspace of V1× · · · × V_d and let U be a subspace of W with dim U = n. If dim W > n(d − 1), then there exists (v1, . . . ,vd) ∈W \ U such that vi 6= 0 for every i. When (q, n, d ) /∈ {(3, 1, 2), (2, 2, 2)}, there are at least two F-linearly independent elements satisfying this property.

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COROLLARY

Let G be a finite soluble group with d = d (G) ≥ 2. Suppose that there exist g1, . . . ,gd generating G with the property that there is a unique maximal subgroup M of G with M ∩ {g1, . . . ,gd} = ∅. Then

|G : M| = 2 and every normal subgroup N of G with d(G/N) = d is contained in G⁰G².

This Corollary can be considerably strengthened when d (G) = 2. COROLLARY

Let G be a finite group with d (G) = 2. Suppose that there exist g₁,g₂ generating G with the property that there is a unique maximal subgroup M of G with M ∩ {g₁,g₂} = ∅. Then |G : M| = 2, G is nilpotent and O2⁰(G) is cyclic.

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COROLLARY

This Corollary can be considerably strengthened when d (G) = 2.

COROLLARY

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COROLLARY

This Corollary can be considerably strengthened when d (G) = 2.

COROLLARY

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A FOLKLORE RESULT

Assume thatG is a soluble primitive permutation groupon a finite set Ω with d (G) = 2. Observe that G = V o H and that M_V = {Gω| ω ∈ Ω}, where Gωis the stabilizer of ω ∈ Ω.

We denote bysupp(g)the support {ω ∈ Ω | ω^g 6= ω} of the permutation g.If G = hg1,g2i, then supp(g₁) ∩supp(g2) 6= ∅.

{M ∈ M_V | M ∩ {g₁,g2} = ∅} = {Gω| Gω∩ {g₁,g2} = ∅}

= {Gω| ω ∈ supp(g₁) ∩supp(g2)}

hence the number of maximal subgroups M ∈ MV avoiding {g₁,g2} is exactly | supp(g₁) ∩supp(g2)|.

When | supp(g₁) ∩supp(g₂)| =1, we have a unique choice for M and, by the previous theorem, either G ∼= Sym(3) or G ∼= Sym(4).

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A FOLKLORE RESULT

If G = hg1,g2i is a soluble primitive permutation group and

| supp(g₁) ∩supp(g2)| =1, then either G ∼= Sym(3) or G ∼= Sym(4).

This has a rather remarkable application. Fix n ∈ N and a ∈ {2, . . . , n − 1}.

Consider the two cycles g₁= (1, . . . , a) and g₂= (a + 1, . . . , n), G = hg1,g2i is a primitive subgroup of Sym(n).

Since supp(g₁) ∩supp(g₂) = {a}, we deduce that either n ≤ 4 or G is insoluble.

In this way we prove that Sym(n) is insoluble for n ≥ 5 using an argument that relies only on linear algebra.

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A FOLKLORE RESULT

This has a rather remarkable application.

Fix n ∈ N and a ∈ {2, . . . , n − 1}.

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A FOLKLORE RESULT

This has a rather remarkable application.

Fix n ∈ N and a ∈ {2, . . . , n − 1}.

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R EMARK 1

LetW := {V ∈ V | δ_G(V ) ≥ (d − 1 − θ_G(V ))n_V+1}. Our theorem states that the condition V ∈ W is sufficient to ensure the existence of a maximal subgroup in MV avoiding g1, . . . ,gd.

QUESTION

IsWa non-empty set?

Let N be the set of normal subgroups N of G with d (G/N) = d and d (G/K ) < d whenever N < K E G.

Let N ∈ N , let K /N be an arbitrary minimal normal subgroup of G/N and let V = K /N. It follows easily that V ∈ V.

The irreducible G-module V satisfies:

(I) δ_G(V ) ≥ (d (G) − 1 − θ_G(V ))n_V+1 (II) d (G/C_G(V ) < d (G).

In other words, for each N ∈ N , the minimal normal subgroups of G/N give rise to irreducible G-modules V ∈ W.

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R EMARK 1

QUESTION

IsWa non-empty set?

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R EMARK 1

QUESTION

IsWa non-empty set?

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R EMARK 2

Observe that d (G/C_G(V )) ≤ d (G) = d . Whend (G/C_G(V )) < d , the condition δ_G(V ) ≥ (d − 1 − θ_G(V ))n_V+1 is necessary and sufficient to ensure that, for every generating d -tuple g1, . . . ,gd, there exists M ∈ MV with M ∩ {g1, . . . ,gd} = ∅.

Indeed, if δG(V ) ≤ (d − 1 − θG(V ))nV and d (G/CG(V )) < d , then d (G/RG(V )) ≤ d − 1 and hence there exist x1, . . . ,xd −1∈ G with G = hx1, . . . ,xd −1,RG(V )i. By a result of Gaschütz, there exist r1, . . . ,rd ∈ R_G(V ) with G = hx1r1, . . . ,xd −1rd −1,rdi: since R_G(V ) =T

M∈MVM, we have r_d ∈ M ∩ {x₁r₁, . . . ,x_{d −1}r_{d −1},r_d} for every M ∈ M_V.

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R EMARK 2

Observe that d (G/C_G(V )) ≤ d (G) = d . Whend (G/C_G(V )) < d , the condition δ_G(V ) ≥ (d − 1 − θ_G(V ))n_V+1 is necessary and sufficient to ensure that, for every generating d -tuple g1, . . . ,gd, there exists M ∈ MV with M ∩ {g1, . . . ,gd} = ∅.

Indeed, if δ_G(V ) ≤ (d − 1 − θ_G(V ))nV and d (G/C_G(V )) < d , then d (G/RG(V )) ≤ d − 1 and hence there exist x1, . . . ,xd −1∈ G with G = hx1, . . . ,xd −1,RG(V )i. By a result of Gaschütz, there exist r1, . . . ,rd ∈ R_G(V ) with G = hx1r1, . . . ,xd −1rd −1,rdi: since RG(V ) =T

M∈MVM, we have rd ∈ M ∩ {x₁r1, . . . ,xd −1rd −1,rd} for every M ∈ M_V.

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R EMARK 3

The condition δ_G(V ) ≥ (d − 1 − θ_G(V ))nV+1 in general is not necessary when d (G/C_G(V )) = d .Indeed, let ˜G be the soluble primitive permutation group VoG/CG(V ): d ( ˜G) = d and a sufficient condition for the existence of M ∈ M_V with M ∩ {g₁, . . . ,g_d} = ∅ is that ∩_1≤i≤dsupp(˜g_i) 6= ∅. This always holds true when d = 3 :

THEOREM

If G = hg1,g2,g3i is a primitive group with d (G) = 3, then supp(g1) ∩supp(g2) ∩supp(g3) 6= ∅.

In particular, when d (G) = d (G/CG(V )) = 3, there always exists M ∈ MV with M ∩ {g1,g2,g3} = ∅, regardless of whether the condition δG(V ) ≥ (d − 1 − θG(V ))nV+1 holds or not.

We do not have any example of a finite soluble group G = hg₁, . . . ,g_di with d = d (G) = d (G/C_G(V )) and of a non-trivial G-module V ∈ V where there is no M ∈ M_V with M ∩ {g₁, . . . ,g_d} = ∅.

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R EMARK 3

The condition δ_G(V ) ≥ (d − 1 − θ_G(V ))nV+1 in general is not necessary when d (G/C_G(V )) = d .Indeed, let ˜G be the soluble primitive permutation group VoG/CG(V ): d ( ˜G) = d and a sufficient condition for the existence of M ∈ M_V with M ∩ {g₁, . . . ,g_d} = ∅ is that ∩_1≤i≤dsupp(˜g_i) 6= ∅. This always holds true when d = 3 : THEOREM

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R EMARK 3

The condition δ_G(V ) ≥ (d − 1 − θ_G(V ))nV+1 in general is not necessary when d (G/C_G(V )) = d .Indeed, let ˜G be the soluble primitive permutation group VoG/CG(V ): d ( ˜G) = d and a sufficient condition for the existence of M ∈ M_V with M ∩ {g₁, . . . ,g_d} = ∅ is that ∩_1≤i≤dsupp(˜g_i) 6= ∅. This always holds true when d = 3 : THEOREM

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QUESTION

Let G be a primitive permutation group and let d = d (G). Does G = hg₁, . . . ,g_di impliesT

1≤i≤dsupp(g_i) = ∅?

A weaker result in this direction is the following: THEOREM

If G = hg1, . . . ,gdi is a primitive permutation group with d (G) = d, then supp(g_i) ∩supp(g_j) 6= ∅for all i, j ∈ {1, . . . , d }.

This does not remain true if we replace “primitive” with “transitive”. Takeg₁= (1, 2, 3, 4), g₂= (5, 7), g₃= (1, 5)(2, 6)(3, 7)(4, 8): G = hg₁,g₂,g₃i is a Sylow 2-subgroup of Sym(8): in particular d (G) = 3 but supp(g₁) ∩supp(g₂) = ∅.

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QUESTION

Let G be a primitive permutation group and let d = d (G). Does G = hg₁, . . . ,g_di impliesT

1≤i≤dsupp(g_i) = ∅?

A weaker result in this direction is the following:

THEOREM

If G = hg1, . . . ,gdi is a primitive permutation group with d (G) = d, then supp(gi) ∩supp(gj) 6= ∅for all i, j ∈ {1, . . . , d }.

This does not remain true if we replace “primitive” with “transitive”.

Takeg₁= (1, 2, 3, 4), g₂= (5, 7), g₃= (1, 5)(2, 6)(3, 7)(4, 8):

G = hg₁,g₂,g₃i is a Sylow 2-subgroup of Sym(8): in particular d (G) = 3 but supp(g₁) ∩supp(g₂) = ∅.

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THEOREM

If G = hg1, . . . ,gdi is a primitive permutation group with d (G) = d, then supp(gi) ∩supp(gj) 6= ∅for all i, j ∈ {1, . . . , d }.

Ingredients of the proof:

O’Nan-Scott theorem.

J. McLaughlin’s classification of irreducible finite dimensional linear groups generated by transvections over the field of 2 elements (1969).

Classification of finite primitive groups admitting a non-identity element fixing at least half of the points of the domain (R.

Guralnick and K. Magaard, 1998).

If N is the unique minimal normal subgroup of a finite group G then d (G) ≤ max{2, d (G/N)} (AL, F. Menegazzo, 1997).

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D IRECT PRODUCT OF NON - ABELIAN SIMPLE GROUPS

LetSbe a finite non-abelian simple group. Given a positive integer d ≥ 3, consider the action of Aut(S) on S^d and letΩ_d be the set of Aut(S)-orbits on the set of d -tuples (x₁, . . . ,x_d) ∈S^d such that:

S = hx1, . . . ,xdi;

for every maximal subgroup M of S, there exists i with x_i ∈ M.

We use the notation [(x1, . . . ,xd)]to denote the Aut(S)-orbit containing (x₁, . . . ,x_d) ∈ Ω_d.We define the graphΓ_dwith vertex set Ω_d and where two distinct vertices [(x₁, . . . ,x_d)]and [(y₁, . . . ,y_d)]are declared to be adjacent if and only if, for every γ ∈ Aut(S), there exists i ∈ {1, . . . , d } (which may depend on γ) such that y_i =x_i^γ. THEOREM

Let ω(Γd)be the clique number of Γd and let P_S(k ) be the probability of generating S with k -elements. We have

ω(Γ_d) ≤ PS(d − 1)|S|^{d −1}

| Aut(S)| .