Minimally irreducible groups of degree a product of two primes

Francesca Dalla Volta  Lino Di Martino

Andrea Previtali

Donnerstag, 29. November 2001 Definition 1. Let χ be a character afforded by a faithful ℂG-module V. We say G is χ-minimally irreducible if G acts irreducibly on V and none of its proper subgroups does.

Aim

Classify minimally irreducible groups G of degree a product of two primes p and q admitting a minimal normal non-abelian subgroup N ≠ G.

LeitFaden

1. Minimally irreducible groups of prime degree;

2. N ·⊲ G, simple, χ_{↾N} = ∑_{1}^{q} θ_{i};

3. N ·⊲ G, semisimple, χ_{↾N} = ∑_{1}^{q} θ_{i};

4. N ·⊲ G, simple, χ_{↾N} = qθ.

General Remarks

Lemma 2. If • G χ-minimally irreducible, N ⊴ G;

• χ_{↾N} = ∑_{1}^{q} θ_{i}, q prime. Then G/N is a cyclic q-group. Lemma 3. G χ-minimally irreducible, χ(1) prime. Then G is

• metabelian;

• simple non-abelian.

Suprunenko groups

Theorem 4 (Suprunenko ’70). G minimally irreducible soluble of prime degree p

IFF

Z(G) cyclic and

1. |G| = p^{3} extraspecial;

2. G = ⟨a, b | a^{p^{n}} = b^{p} = 1, [a, b] = a^{p^{n−1}}⟩, modular;

3. V : X, X p-cyclic, V irreducible X/℧_{1}(X)-module.

The last class generalizes the affine linear groups in dimension 1.

Technical results

Lemma 5. For S in LSZ, π(Out(S)) ⊆ {2, n}. Hence p ∤ |Out(S)|, whenever Irr_{p}(S) ≠ ∅. Proposition 6. If

• G = NY, χ-irreducible, χ(1) = pq;

• N normal, χ_{↾N} = ∑_{1}^{q} θ_{i};

• Y cyclic q-group;

• M {< \atop ·} G irreducible;

• H = M ∩ N. Then

1. θ_{↾H} ∈ Irr_{p}(H) or

2. p = q and H abelian.

Proposition 7. If H ≤ N irreducible such that: • χ_{↾H} = ∑_{1}^{q} η_{i};

• N_{G}(H) ≰ I_{G}(θ). Then G ≥ M = HX irreducible, N_{G}(H) ≥ X q-cyclic. Proposition 8. If

• G = NY, N ⊴ G;

• Y q-cyclic, C = C_{G}(N);

• ∃θ ∈ Irr_{p}(N), γ ∈ Irr_{1}(C) s. t. |(θγ)^{Y} | = q;

• Core_{G}(ker(θγ)) = 1. Then ∃χ ∈ Irr_{pq}(G) faithful.

The Non-homogeneous Case

Assume

1. χ ∈ Irr_{pq}(G) faithful;

2. N ·⊲ G non-abelian;

3. χ_{↾N} = ∑_{I}^{q} θ_{i}. Then

• N is simple non-abelian;

• N = ⧹ ⧸_{i}^{q}S_{i}, S_{i} ≃ S simple non-abelian.

N simple

𝒜: G = SY, χ-irreducible, S simple, Y q-cyclic, χ_{↾S} = ∑_{1}^{q} θ_{i}, θ_{i} ∈ Irr_{p}(S). Lemma 9. If G ∈ 𝒜, then {θ_{i}} is an orbit in Irr_{p}(S) under some cyclic subgroup of Out(S). In particular q | |Out(S)|.

Orbit Problem

If S = PSL_{n}(u), p = \frac{u^{n}−1}{u−1}, u = r^{f}, r prime, j ∼ i iff j ≡ ±i (mod u − 1). Then (Irr_{p}(S), Fr) ≈ (Ω, μ_{r}), where

• Ω = ℤ_{u−1}^{∗}, for u odd;

• Ω = ℤ_{u−1}^{∗}/ ∼, for u even.

Theorem 10. If • S in LSZ, θ ∈ Irr_{p}(S);

• 1 ≠ |θ^{X}|, Out(S) ≥ X, q-cyclic. Then (S, p, |θ^{X}|):

1. (PSL_{2}(u), \frac{u±1}{2}, 2);

2. (PSL_{2}(u), u − 1, q), u = 2^{q};

3. (PSL_{n}(u), \frac{u^{n}−1}{u−1}, n^{t}), n = q ≥ 2 < u, and (q, u) ≠ (2, 4);

4. (PSL_{n}(u), \frac{u^{n}−1}{u−1}, 2), 2 ≠ n prime, u > 2;

5. (PSp_{2n}(u), \frac{u^{n}±1}{2}, 2);

6. (M_{12}, 11, 2).

When G ∈ 𝒜 is minimally irreducible?

IFF

No maximal irreducible subgroups M as in Proposition ??.

IFF

H = M ∩ S neither irreducible nor abelian (p = q here).

By Lemma ??, H can not be abelian. Lemma 11 (Symplectic embeddings). If

• F = 𝔽_{u}, [E : F] = d;

• R : E → (F)_{d} regular representation.

Then ∃t ∈ GL_{d}(u) such that R(x)^{t} = R(x)^{u} and tt′ = 1.

Theorem 12. Assume • G ⊵ S simple, C = C_{G}(S);

• θ ∈ Irr_{p}(S), |θ^{X}| = q, X ≤ Out(S). G is χ-minimal, χ_{↾S} = ∑_{i=1}^{q} θ_{i}

IFF

1. G = SY, Y q-cyclic, {θ_{i}} a Y -orbit;

2. (S, p, q) is one of the following:

(a) (PSL_{2}(u), \frac{u+1}{2}, 2);

(b) (PSL_{2}(2^{q}), 2^{q} − 1, q);

(c) (PSL_{q}(u), \frac{u^{q}−1}{u−1}, q), q ≥ 2, u > 2;

(d) (PSL_{n}(u), \frac{u^{n}−1}{u−1}, 2), n ≠ 2 prime, u > 2;

(e) (PSp_{2n}(u), \frac{u^{n}+1}{2}, 2), (n, u) ≠ (2, 3);

(f) (M_{12}, 11, 2).

3. Any (linear) constituent of χ_{↾C} is faithful;

4. [C, ℧_{1}(Y)] = 1.

Examples

• S non-minimally irreducible: S = Alt_{5} and G = Sym_{5} of degree 6;

• S minimally irreducible: G = PSL_{2}(16) : ℤ_{2} of degree 34.

Structure of CY

Proposition 13. Assume • C ⊴ CY, C and Y q-cyclic;

• [C, ℧_{1}(Y)] = 1 and |C| < |Y |. Then CY is one of the following metacyclic groups:

1. cyclic: ⟨a | a^{q^{l}} = 1⟩, l ≥ 1;

2. abelian: ⟨a, b | a^{q^{l}} = b^{q^{m}} = [a, b] = 1⟩, m > l ≥ 1;

3. generalized modular:

⟨a, b | a^{q^{l}} = b^{q^{m}} = 1, a^{q^{l−1}} = [a, b]⟩, 1 < l ≠ m;

4. generalized dihedral:

⟨a, b | a^{2^{l}} = b^{2^{m}} = 1, a^{b} = a^{−1}⟩, 1 < l < m;

5. generalized semidihedral:

⟨a, b | a^{2^{l}} = b^{2^{m}} = 1, a^{b} = a^{−1+2^{l−1}}⟩, 2 < l < m;

6. non-split semidihedral:

⟨a, b | a^{2^{l}} = b^{2^{m}}, a^{2^{l+1}} = 1, a^{b} = a^{−1+2^{l}}⟩, 1 < l < m.

Do all cases occur? At least CY may be non-cyclic. Example 14 (Meierfrankenfeld). Let S = PSL_{2}(9) and G = M_{10} = S⟨DiFr⟩, then C_{G}(S) is never a subgroup of a cyclic supplement to S in G.

N = S^{q}

ℬ: G = NY, χ-irreducible, N = S^{q}, S simple, Y q-cyclic, χ_{↾N} = ∑_{1}^{q} θ_{i}, θ_{i} ∈ Irr_{p}(N).

• H = {(s^{α_{1}}, …, s^{α_{q}})}, s ∈ S, α_{i} ∈ Aut(S) is called diagonal;

• H = ⧹ ⧸_{i}T_{i} is a product subgroup of S^{q}, T_{i} ≤ S_{i}.

Facts-Notation

1. Y = ⟨(ω_{1}, …, ω_{q})βγ⟩, where ω_{i} ∈ Aut(S), β = (1, …, q) ∈ Sym_{q}, and γ ∈ C_{G}(N);

2. Irr(N) ∋ θ = σ × 1 × … × 1, σ ∈ Irr_{p}(S);

3. Set Ω_{i} = ω_{1} ⋯ ω_{i}.

S not σ-minimal

Proposition 15. Assume • T ≤ S is σ-irreducible, Ω_{q}-invariant;

• T_{i+1} = T^{Ω_{i}} and H = ⧹ ⧸_{i}T_{i}. Then HY is a χ-irreducible subgroup of G. Theorem 16. If G ∈ ℬ, S not σ-minimal. Then G is minimally irreducible iff S = PSL_{2}(9), q = 2 and Ω_{2} ∈ δInn(S), δ = Di or DiFr.

Example

G = (PSL_{2}(9) × PSL_{2}(9))⟨(1, Di)β⟩, β acts as a 2-cycle on the direct factors.

S σ-minimal

Theorem 17. Let G ∈ ℬ, S σ-minimally irreducible, M ≤ G irreducible. Then H = N ∩ M is not abelian. Definition 18. N ≥ H = ⧹ ⧸D_{i}, with D_{i} diagonal, is a multidiagonal subgroup. Lemma 19. Let H ≤ S^{q}, S simple non-abelian. Then H^{π_{i}} = S_{i} and H ∩ S_{i} = 1 iff H is a multidiagonal thick subgroup.

Theorem 20. Assume • S minimally irreducible;

• M a proper irreducible subgroup of G;

• H = M ∩ N non-abelian;

• χ_{↾H} = ∑_{1}^{q} η_{i}. Then H is diagonal. In particular H lives in LSZ Table, η ∈ Irr_{p}(H). Theorem 21. Let G ∈ ℬ, S σ-minimal. Then G is minimally irreducible unless (S, p, q):

1. (PSL_{2}(u), \frac{u+1}{2}, 2), \overline{Ω_{2}} ∈ ℧_{1}(⟨\overline{Fr}⟩);

2. (PSL_{2}(u), u + 1, 2), ⟨\overline{Ω_{2}}⟩ = \overline{I(η)};

3. (PSL_{n}(u), \frac{u^{n}−1}{u−1}, 2), 2 ≠ n prime, η ≠ θ_{\frac{u−1}{2}} ;

4. (PSL_{n}(u), \frac{u^{n}−1}{u−1}, n), 2 ≠ n prime, η ≠ θ_{\frac{u−1}{2}} , and ⟨\overline{Ω_{n}}⟩ = \overline{I(η)}.

The Homogeneous Case

Lemma 22. Let G possess an irreducible faithful character χ. Assume N ·⊲ G such that χ_{↾N} = qθ, θ ∈ Irr(N), then N is simple. 𝒞: G, χ-irreducible, G ⊴ S simple, χ_{↾S} = qθ, θ ∈ Irr_{p}(S). Theorem 23. Assume

• M, N ·⊲ G, N ≠ M non-abelian;

• χ_{↾N} is homogeneous. Then N, M are simple and G = N × M.

Direct Products

Theorem 24. Let G = ⧹ ⧸_{1}^{r}N_{i}, r ≥ 2. Then G is χ-minimal, χ(1) = ∏_{i=1}^{r} p_{i}, p_{i} prime,

IFF

N_{i} is θ_{i}-minimal, θ_{i}(1) = p_{i}, and gcd(|Z(N_{i})|, |Z(N_{j})|) = 1. We may assume G ≠ S × C, C = C_{G}(S). Proposition 25. If 𝒞 ∋ G ≠ S × C, χ-minimal, χ(1) = pq. Then

1. C is abelian;

2. χ_{↾SC} = θ ∑_{1}^{q} γ_{i}, γ_{i} ∈ Irr_{1}(C);

3. G/SC is q-cyclic.

Theorem 26. Assume 𝒞 ∋ G ≠ S × C. Then G is χ-minimally irreducible of degree pq, with χ_{↾SC} = θ ∑_{1}^{q} γ_{i} iff

1. 1 ≠ G/SC ≤ I_{Out(S)}(θ), q-cyclic;

2. S is θ-minimally irreducible and (S, p, q): (a) (PSL_{2}(r), r, 2), θ = St;

(b) (PSL_{2}(2^{2^{s}}), 2^{2^{s}} + 1, 2);

(c) (PSL_{2}(r^{2^{s}}), \frac{r^{2^{s}} +1}{2}, 2);

(d) (PSL_{n}(u), \frac{u^{n}−1}{u−1}, n), n ≠ 2 prime;

(e) (PSL_{n}(u), \frac{u^{n}−1}{u−1}, 2), θ = θ_{\frac{u−1}{2}} ;

(f) (PSU_{n}(u), \frac{u^{n}+1}{u+1}, n), n ≠ 2 prime;

(g) (PSU_{n}(u), \frac{u^{n}+1}{u+1}, 2).

3. G = S : CY, Y q-cyclic;

4. {γ_{i}} = γ^{Y} and ⋂_{i} ker γ_{i} = 1, in particular C is abelian, but CY is not;

5. C_{C}(Y) is a q-group and [T, Y] ≤ T < C ⇒ [T, Y] = 1.

Definition 27. CY is almost-Suprunenko if • C is abelian, Y q-cyclic;

• ∃γ ∈ Irr_{1}(C) such that |γ^{Y} | = q and ⋂_{i} ker γ_{i} = 1;

• C_{C}(Y) is a q-group and

[T, Y] ≤ T < C ⇒ [T, Y] = 1. Definition 28. The exotic group is ⟨a, b | a^{q} = b^{q^{l}} = [a, b, b] = [a, b, a] = 1⟩. Theorem 29. Q is almost Suprunenko iff 1. non-nilpotent Suprunenko;

2. generalized modular;

3. exotic;

4. the quaternion group of order 8.

Open Problems

1. Establish the occurrence of CY given in Proposition ?? and Theorem ??;

2. classify minimally irreducible simple groups of degree pq (Deligne-Lusztig Theory);

3. classify minimally irreducible soluble groups of degree pq.