B OUNDEDLY G ENERATED S UBGROUPS OF
F INITE G ROUPS
Andrea Lucchini
Università di Padova, Italy
2nd Biennial International Group Theory Conference Istanbul, February 4-8, 2013
AN(EASY?) QUESTION
Assume that G is a finite group and let π(G) be the set of prime divisors of the order of G. Does there exist a 2-generated subgroup H of G with π(H) = π(G)?
YES!and a stronger result can be proved:
THEOREM(AL, M. MORIGI, P. SHUMYATSKY(2011))
Let C(G) be the set of isomorphism classes of composition factors of G, then there exists a 2-generated subgroup H of G such that C(H) = C(G).
AN(EASY?) QUESTION
Assume that G is a finite group and let π(G) be the set of prime divisors of the order of G. Does there exist a 2-generated subgroup H of G with π(H) = π(G)?
YES!and a stronger result can be proved:
THEOREM(AL, M. MORIGI, P. SHUMYATSKY(2011))
Let C(G) be the set of isomorphism classes of composition factors of G, then there exists a 2-generated subgroup H of G such that C(H) = C(G).
AN(EASY?) QUESTION
Assume that G is a finite group and let π(G) be the set of prime divisors of the order of G. Does there exist a 2-generated subgroup H of G with π(H) = π(G)?
YES!and a stronger result can be proved:
THEOREM(AL, M. MORIGI, P. SHUMYATSKY(2011))
Let C(G) be the set of isomorphism classes of composition factors of G, then there exists a 2-generated subgroup H of G such that C(H) = C(G).
H OW A RESULT LIKE THIS CAN BE PROVED ?
Let L be aprimitive monolithic group(L has a unique minimal normal subgroup A, and if A is abelian then A has a complement in G).
Thecrown-based powerof L of size k is the subgroup Lk of Lk defined by:Lk = {(l1, . . . ,lk) ∈Lk|l1≡ . . . ≡ lk mod A}.
Let d (G) denotes the minimal number of generators of the group G.
THEOREM(F. DALLAVOLTA, AL (1998))
Let m be a natural number and let G be a finite group such that d (G/N) ≤ m for every non-trivial normal subgroup N, but d (G) > m.
Then there exists a primitive monolithic group L such that G ∼= Lk for some k .
LetH ≤ G, minimal with the property thatC(H) = C(G).
⇓
There existsNE H such that:d (H/N) = d (H)butd (H/M) < d (H) whenever N < M E H.
⇓
There existLandk withH/N ∼= Lk.
⇓
There exists N ≤ H∗ ≤ H with H∗/N ∼= L. Since C(H) = C(H∗), it must be H = H∗, hencek = 1.
THEOREM(AL, F. MENEGAZZO(1997))
If L is a primitive monolithic group, then d (L) = max(2, d (L/ soc L)). Let M/N = soc H/N :d (H) = d (H/N) = max{2, d (H/M)} = 2.
LetH ≤ G, minimal with the property thatC(H) = C(G).
⇓
There existsNE H such that:d (H/N) = d (H)butd (H/M) < d (H) whenever N < M E H.
⇓
There existLandk withH/N ∼= Lk.
⇓
There exists N ≤ H∗ ≤ H with H∗/N ∼= L. Since C(H) = C(H∗), it must be H = H∗, hencek = 1.
THEOREM(AL, F. MENEGAZZO(1997))
If L is a primitive monolithic group, then d (L) = max(2, d (L/ soc L)). Let M/N = soc H/N :d (H) = d (H/N) = max{2, d (H/M)} = 2.
LetH ≤ G, minimal with the property thatC(H) = C(G).
⇓
There existsNE H such that:d (H/N) = d (H)butd (H/M) < d (H) whenever N < M E H.
⇓
There existLandk withH/N ∼= Lk.
⇓
There exists N ≤ H∗ ≤ H with H∗/N ∼= L. Since C(H) = C(H∗), it must be H = H∗, hencek = 1.
THEOREM(AL, F. MENEGAZZO(1997))
If L is a primitive monolithic group, then d (L) = max(2, d (L/ soc L)). Let M/N = soc H/N :d (H) = d (H/N) = max{2, d (H/M)} = 2.
LetH ≤ G, minimal with the property thatC(H) = C(G).
⇓
There existsNE H such that:d (H/N) = d (H)butd (H/M) < d (H) whenever N < M E H.
⇓
There existLandk withH/N ∼= Lk.
⇓
There exists N ≤ H∗≤ H with H∗/N ∼= L.
Since C(H) = C(H∗), it must be H = H∗, hencek = 1.
THEOREM(AL, F. MENEGAZZO(1997))
If L is a primitive monolithic group, then d (L) = max(2, d (L/ soc L)). Let M/N = soc H/N :d (H) = d (H/N) = max{2, d (H/M)} = 2.
LetH ≤ G, minimal with the property thatC(H) = C(G).
⇓
There existsNE H such that:d (H/N) = d (H)butd (H/M) < d (H) whenever N < M E H.
⇓
There existLandk withH/N ∼= Lk.
⇓
There exists N ≤ H∗≤ H with H∗/N ∼= L.
Since C(H) = C(H∗), it must be H = H∗, hencek = 1.
THEOREM(AL, F. MENEGAZZO(1997))
If L is a primitive monolithic group, then d (L) = max(2, d (L/ soc L)).
Let M/N = soc H/N :d (H) = d (H/N) = max{2, d (H/M)} = 2.
LetH ≤ G, minimal with the property thatC(H) = C(G).
⇓
There existsNE H such that:d (H/N) = d (H)butd (H/M) < d (H) whenever N < M E H.
⇓
There existLandk withH/N ∼= Lk.
⇓
There exists N ≤ H∗≤ H with H∗/N ∼= L.
Since C(H) = C(H∗), it must be H = H∗, hencek = 1.
THEOREM(AL, F. MENEGAZZO(1997))
If L is a primitive monolithic group, then d (L) = max(2, d (L/ soc L)).
Let M/N = soc H/N :d (H) = d (H/N) = max{2, d (H/M)} = 2.
The previous proof suggests a strategy to tackle other similar (more difficult) questions:
QUESTION
Let i(G) be a group invariant. Does there exist d ∈ N such that every finite group G contains a d -generated subgroup H with i(H) = i(G)?
STRATEGY
Let H ≤ G minimal such that i(H) = i(G) and let d = d (H).
There is N E H s.t. d(H/N) = d, d(H/M) < d ∀N < M E H.
H/N ∼= Lk for suitable L and k .
Can we prove that k ≤ t, where t depends on the invariant we are considering but not on H?
If the previous question has a positive answer andwe can prove that there exists an absolute constant τ such that
d (Lt) ≤max(d (L), τ ) for any choice of L, then we can conclude that d (H) ≤ τ.
d (L
k) ≤ f (k )
Let L be a primitive monolithic group, A = soc L. Assume d (L) ≤ d .
ASSUME THATAIS ABELIAN
Let q = | EndL/A(A)|, qr = |A|, qs = |H1(L/A, A)|, θ = 0 or 1 according to whether A is a trivial L/A-module or not. Then
d (Lk) ≤d ⇔ k ≤ r (d − θ) − s.
Notice that s < r , as a consequence of a result of Aschbacher and Guralnick, indeed we have:|H1(L/A, A)| < |A|.
In particular d (Lk) ≤max(d (L), d(k + s)/r e + θ) ≤ max(d (L), k + 1).
d (L
k) ≤ f (k )
Let L be a primitive monolithic group, A = soc L. Assume d (L) ≤ d .
ASSUME THATAIS NON ABELIAN
LetPL,A(d )be the conditional probability that d randomly chosen elements of L generate L given that they generate L modulo A. Then
d (Lk) ≤d ⇔ k ≤ PL,A(d )|A|d
|CAut(A)(L/A)|.
A = Snwhere n is a positive integer and S is a nonabelian simple group. It is not difficult to prove that |CAut(A)(L/A)| ≤ n|S|n−1| Aut(S)|.
We need an estimation of PL,A(d ).
d (L
k) ≤ f (k )
Let L be a primitive monolithic group with non-abelian socle A ∼= Sn. THEOREM(E. DETOMI, C. RONEY-DOUGAL, AL (2013))
If d ≥ d (L) then there exists a 2-generated almost simple group Y with socle S such that PL,A(d ) ≥ PY ,S(2).
THEOREM(N. MENEZES, M. QUICK, C. RONEY-DOUGAL(2012)) If Y is a 2-generated almost simple group with non abelian socle S, then PY ,S(2) ≥ 53/90 with equality if and only if Y = Alt(6), Sym(6).
It follows:if d (L) ≥ d , then PL,A(d ) ≥ 53/90.
MoreoverPL,A(d ) → 1 as |A| → ∞.
P RIME GRAPH
Denote by Γ(G) theprime graphof a finite group G. This is the graph whose set of vertices is π(G) and p, q ∈ π(G), with p 6= q, are connected by an edge if and only if G has an element of order pq.
THEOREM(M. MORIGI, P. SHUMYATSKY, AL (2011))
Let G be a finite group. Then there exists a 3-generated subgroup H of G such that Γ(H) = Γ(G).
The previous theorem cannon be improved.
P RIME GRAPH
Denote by Γ(G) theprime graphof a finite group G. This is the graph whose set of vertices is π(G) and p, q ∈ π(G), with p 6= q, are connected by an edge if and only if G has an element of order pq.
THEOREM(M. MORIGI, P. SHUMYATSKY, AL (2011))
Let G be a finite group. Then there exists a 3-generated subgroup H of G such that Γ(H) = Γ(G).
The previous theorem cannon be improved.
P RIME GRAPH
Denote by Γ(G) theprime graphof a finite group G. This is the graph whose set of vertices is π(G) and p, q ∈ π(G), with p 6= q, are connected by an edge if and only if G has an element of order pq.
THEOREM(M. MORIGI, P. SHUMYATSKY, AL (2011))
Let G be a finite group. Then there exists a 3-generated subgroup H of G such that Γ(H) = Γ(G).
The previous theorem cannon be improved.
V1= F5× F5, V2= F7× F7. Sym(3) ∼=
α1:=1 1 0 −1
, β1:=1 1 2 3
≤ GL(V1).
Sym(3) ∼=
α2:=0 1 1 0
, β2:= 0 1
−1 −1
≤ GL(V2).
(Sym(3))2∼= K = h(β1,1), (1, β2), (α1, α2)i ≤GL(V1) ×GL(V2).
G:= (V1× V2) o K .
d (G) = 3, but Γ(H) 6= Γ(G) for all H < G.
s s
s
s 15||g| ⇒ g ∈ V hβ2i 21||g| ⇒ g ∈ V hβ1i 5
2
3 7
V1= F5× F5, V2= F7× F7. Sym(3) ∼=
α1:=1 1 0 −1
, β1:=1 1 2 3
≤ GL(V1).
Sym(3) ∼=
α2:=0 1 1 0
, β2:= 0 1
−1 −1
≤ GL(V2).
(Sym(3))2∼= K = h(β1,1), (1, β2), (α1, α2)i ≤GL(V1) ×GL(V2).
G:= (V1× V2) o K .
d (G) = 3, but Γ(H) 6= Γ(G) for all H < G.
s s
s
s 15||g| ⇒ g ∈ V hβ2i 21||g| ⇒ g ∈ V hβ1i 5
2
3 7
S PECTRUM
Thespectrumof a group G is the set of orders of the elements of G.
For every positive integer d there exists a group G such that d (G) = d and no (d − 1)-generated subgroup of G has the same spectrum.
X = {p1, . . . ,pd} a set of d different odd prime numbers Di = hbi,ai | bpii =a2i =1, baii =bi−1i ∼= D2pi, G =Qd
i=1Di
if d ≥ 2, then d (G) = d
the spectrum of G is the set of the proper divisors of m = 2p1· · ·pd
the elements of order m/pi are of the form (br11, . . . ,bri−1i−1,aibrii,bi+1ri+1, . . . ,bdrd).
S PECTRUM
Thespectrumof a group G is the set of orders of the elements of G.
For every positive integer d there exists a group G such that d (G) = d and no (d − 1)-generated subgroup of G has the same spectrum.
X = {p1, . . . ,pd} a set of d different odd prime numbers Di = hbi,ai | bpii =a2i =1, baii =bi−1i ∼= D2pi, G =Qd
i=1Di
if d ≥ 2, then d (G) = d
the spectrum of G is the set of the proper divisors of m = 2p1· · ·pd
the elements of order m/pi are of the form (br11, . . . ,bri−1i−1,aibrii,bi+1ri+1, . . . ,bdrd).
S PECTRUM
Thespectrumof a group G is the set of orders of the elements of G.
For every positive integer d there exists a group G such that d (G) = d and no (d − 1)-generated subgroup of G has the same spectrum.
X = {p1, . . . ,pd} a set of d different odd prime numbers Di = hbi,ai | bpii =a2i =1, baii =bi−1i ∼= D2pi, G =Qd
i=1Di
if d ≥ 2, then d (G) = d
the spectrum of G is the set of the proper divisors of m = 2p1· · ·pd
the elements of order m/pi are of the form (br11, . . . ,bri−1i−1,aibrii,bi+1ri+1, . . . ,bdrd).
COMPLEX IRREDUCIBLE CHARACTER DEGREES
Letπcd(G)be the set of the prime divisors of the complex irreducible character degrees of G.
THEOREM
Let G be a finite group. There exists a 3-generated subgroup H of G such that πcd(H) = πcd(G).
The previous theorem cannot be improved.
The non abelian groupPof order 27 and exponent 3 admits an automorphism α of order 2, acting on P/ Frat(P) as the inverting automorphism.
LetG=P o hαi : d(G) = 3 and πcd(G) = {2, 3}.
Let H < G : πcd(H) = {3} if H ≤ P, πcd(H) = {2} otherwise.
COMPLEX IRREDUCIBLE CHARACTER DEGREES
Letπcd(G)be the set of the prime divisors of the complex irreducible character degrees of G.
THEOREM
Let G be a finite group. There exists a 3-generated subgroup H of G such that πcd(H) = πcd(G).
The previous theorem cannot be improved.
The non abelian groupPof order 27 and exponent 3 admits an automorphism α of order 2, acting on P/ Frat(P) as the inverting automorphism.
LetG=P o hαi : d(G) = 3 and πcd(G) = {2, 3}.
Let H < G : πcd(H) = {3} if H ≤ P, πcd(H) = {2} otherwise.
COMPLEX IRREDUCIBLE CHARACTER DEGREES
Letπcd(G)be the set of the prime divisors of the complex irreducible character degrees of G.
THEOREM
Let G be a finite group. There exists a 3-generated subgroup H of G such that πcd(H) = πcd(G).
The previous theorem cannot be improved.
The non abelian groupPof order 27 and exponent 3 admits an automorphism α of order 2, acting on P/ Frat(P) as the inverting automorphism.
LetG=P o hαi : d(G) = 3 and πcd(G) = {2, 3}.
Let H < G : πcd(H) = {3} if H ≤ P, πcd(H) = {2} otherwise.
CONJUGACY CLASS SIZES
Letπcs(G)be the set of the prime divisors of the conjugacy class sizes of G.
For every positive integer d there exists a finite group G such that if H ≤ G and πcs(H) = πcs(G) then d (H) ≥ d .
To each σ = {i, j} ⊆ {1, . . . , t}, with i 6= j, we associate a different primepσ.
Let Aσ be a cyclic group of order pσandA =Q
σAσ.
For 1 ≤ i ≤ t, let Ci = hxii be a cyclic group of order 2 and let C =Q
iCi.
We define an action of C on A: xi centralizes Aσ if i /∈ σ, xi acts of Aσas the inverting automorphism otherwise.
LetG = A o C; πcs(G) = π(G) = {2} ∪ {pσ| σ}. H ≤ G and πcs(H) = πcs(G) =⇒ d (H) ≥ log2t.
CONJUGACY CLASS SIZES
Letπcs(G)be the set of the prime divisors of the conjugacy class sizes of G.
For every positive integer d there exists a finite group G such that if H ≤ G and πcs(H) = πcs(G) then d (H) ≥ d .
To each σ = {i, j} ⊆ {1, . . . , t}, with i 6= j, we associate a different primepσ.
Let Aσ be a cyclic group of order pσandA =Q
σAσ.
For 1 ≤ i ≤ t, let Ci = hxii be a cyclic group of order 2 and let C =Q
iCi.
We define an action of C on A: xi centralizes Aσ if i /∈ σ, xi acts of Aσas the inverting automorphism otherwise.
LetG = A o C; πcs(G) = π(G) = {2} ∪ {pσ| σ}. H ≤ G and πcs(H) = πcs(G) =⇒ d (H) ≥ log2t.
CONJUGACY CLASS SIZES
Letπcs(G)be the set of the prime divisors of the conjugacy class sizes of G.
For every positive integer d there exists a finite group G such that if H ≤ G and πcs(H) = πcs(G) then d (H) ≥ d .
To each σ = {i, j} ⊆ {1, . . . , t}, with i 6= j, we associate a different primepσ.
Let Aσ be a cyclic group of order pσandA =Q
σAσ.
For 1 ≤ i ≤ t, let Ci = hxii be a cyclic group of order 2 and let C =Q
iCi.
We define an action of C on A: xi centralizes Aσ if i /∈ σ, xi acts of Aσas the inverting automorphism otherwise.
LetG = A o C; πcs(G) = π(G) = {2} ∪ {pσ| σ}.
H ≤ G and πcs(H) = πcs(G) =⇒ d (H) ≥ log2t.
Denote byΓcd(G)the graph whose set of vertices is πcd(G) and p, q ∈ π(G), with p 6= q, are connected by an edge if and only if p · q divides the degree of an irreducible complex character of G.
OPEN QUESTION
Does there exist a positive integer c with the property that every finite group G contains a c-generated subgroup H with Γcd(H) = Γcd(G)?
Denote byΓcd(G)the graph whose set of vertices is πcd(G) and p, q ∈ π(G), with p 6= q, are connected by an edge if and only if p · q divides the degree of an irreducible complex character of G.
OPEN QUESTION
Does there exist a positive integer c with the property that every finite group G contains a c-generated subgroup H with Γcd(H) = Γcd(G)?
EXPONENT
THEOREM(E. DETOMI, M. MORIGI, P. SHUMYATSKY, AL (2012)) Let G be a finite group. Then there exists a 3-generated subgroup H of G such that H has the same exponent of G.
A finite solvable group G contains a 2-generated subgroup H with the same exponent. We don’t know whether this is true for arbitrary finite groups.
EXPONENT
THEOREM(E. DETOMI, M. MORIGI, P. SHUMYATSKY, AL (2012)) Let G be a finite group. Then there exists a 3-generated subgroup H of G such that H has the same exponent of G.
A finite solvable group G contains a 2-generated subgroup H with the same exponent. We don’t know whether this is true for arbitrary finite groups.
P ROOF IN THE SOLVABLE CASE
Assume that G is a finite solvable group without proper subgroups of the same exponent. There exist:
1 an elementary abelian p-group A
2 an irreducible subgroup H of Aut(A)
3 a positive integer k such that d (G) = d (Ako H) > d (Ak −1o H)
4 an epimorphism φ : G → Ako H.
Let exp(G) = pam with (p, m) = 1. Choose g ∈ G with |g| = pa:there exist a ∈ Ak, h ∈ H with φ(g) = ah. Consider K = φ−1(ha, Hi).
exp(G) = exp(K )⇒G = K ⇒Ako H = ha, Hi
⇒Ak is a cyclic H-module
⇒d (G) = d (Ako H) ≤ 2
PRIME DIVISORS OF INDICES
THEOREM
Let G be a finite group and let X , C be subgroups of G with X ≤ C.
Then there exist a, b ∈ G s.t. π(|G : C|) ⊆ π(|ha, b, X i : C ∩ ha, b, X i|).
s s
s s
C ha, b, X i
s s
@
@
@
@
@
@
@
@
C ∩ ha, b, X i G
X
When X = C we obtain:
COROLLARY
Let G be a finite group and let C be a subgroup of G. Then there exist a, b ∈ G such that π(|G : C|) = π(|ha, b, Ci : C|).
When X is the identity subgroup we obtain:
COROLLARY
Let G be a finite group and let C be a subgroup of G. Then there exist a, b ∈ G such that π(|G : C|) ⊆ π(|ha, bi : C ∩ ha, bi|).
A N APPLICATION
LetIndG(x )be the index in G of the centralizer CG(x ) of x in G.
THEOREM(A.R CAMINA, P. SHUMYATSKY, C. SICA(2010))
IfIndha,b,xi(x ) is a prime-power for every a, b ∈ G, thenIndG(x ) is a prime-power.
This theorem can be restated: let C = CG(x ) and X = hx i; if there is more than one prime dividing |G : C|, then there exist a, b ∈ G such that |ha, b, X i : C ∩ ha, b, X i| is divisible by more than one prime. From our previous result we have a stronger result: there exist a, b ∈ G with π(|G : C|) ⊆ π(|ha, b, X i : C ∩ ha, b, X i|). In other words: THEOREM
For each x ∈ G there exists a and b in G such that π(IndG(x )) ⊆ π(Indhx,a,bi(x )).
A N APPLICATION
LetIndG(x )be the index in G of the centralizer CG(x ) of x in G.
THEOREM(A.R CAMINA, P. SHUMYATSKY, C. SICA(2010))
IfIndha,b,xi(x ) is a prime-power for every a, b ∈ G, thenIndG(x ) is a prime-power.
This theorem can be restated: let C = CG(x ) and X = hx i; if there is more than one prime dividing |G : C|, then there exist a, b ∈ G such that |ha, b, X i : C ∩ ha, b, X i| is divisible by more than one prime.
From our previous result we have a stronger result: there exist a, b ∈ G with π(|G : C|) ⊆ π(|ha, b, X i : C ∩ ha, b, X i|). In other words:
THEOREM
For each x ∈ G there exists a and b in G such that π(IndG(x )) ⊆ π(Indhx,a,bi(x )).
A N APPLICATION
LetIndG(x )be the index in G of the centralizer CG(x ) of x in G.
THEOREM(A.R CAMINA, P. SHUMYATSKY, C. SICA(2010))
IfIndha,b,xi(x ) is a prime-power for every a, b ∈ G, thenIndG(x ) is a prime-power.
This theorem can be restated: let C = CG(x ) and X = hx i; if there is more than one prime dividing |G : C|, then there exist a, b ∈ G such that |ha, b, X i : C ∩ ha, b, X i| is divisible by more than one prime.
From our previous result we have a stronger result: there exist a, b ∈ G with π(|G : C|) ⊆ π(|ha, b, X i : C ∩ ha, b, X i|). In other words:
THEOREM
For each x ∈ G there exists a and b in G such that π(IndG(x )) ⊆ π(Indhx,a,bi(x )).
Given a subset X of G, letdX(G)denote the minimum integer d such that there exist d elements g1, . . . ,gd ∈ G with the property that G = hX , g1, . . . ,gdi.
THEOREM
Let X be a subset of a finite group G and let N be a normal subgroup of G such that N is maximal with the property that dXN(G) = dX(G).
Then there exist a monolithic primitive group L and an isomorphism ϕ :G/N → Lk such that ϕ(X ) ≤ {(l, . . . , l) ∈ Lk | l ∈ L}.
