Conjugacy of Carter subgroups

A. Previtali, M.C. Tamburini

E.P. Vdovin 21 July 2003, Lisboa Let G be a finite group, we say H is a Carter subgroup of G if

• H is nilpotent;

• H is selfnormalizing.

Denote Car(G) the set of Carter subgroups of G. Theorem 1 (Carter 1961). If G is a finite soluble group, then

1. ℰ (existence): Car(G) ≠ ∅;

2. 𝒞 (conjugacy): Car(G) is a conjugacy class.

If G = A_{5}, the alternating group, then Car(G) = ∅. Thus ℰ needs not hold in the universe of finite groups.

If G = H ∗ K, the free product of two non isomorphic nilpotent groups, then H, K ∈ Car(G). Hence 𝒞 does not hold for any group.

The Conjugacy Conjecture

There is strong evidence for the following: Conjecture 2. Conj^{2}=Conjugacy Conjecture: For any finite group G, Car(G) is either empty or a conjugacy class in G.

Let 𝔑 be the class of nilpotent groups. Given G let

Cov_{𝔑}(G) = {H ≤ G | T = T^{𝔑}H},

for any H ≤ T ≤ G, and Proj_{𝔑}(G) the subgroups H of G such that HN/N is maximal among nilpotent subgroups of G/N, for any N ⊴ G. Lemma 1 (Carter 1961). Let G be a soluble finite group, then Cov_{𝒩} (G) = Proj_{𝒩} (G) = Car(G).

From this Lemma follows that Car(G) is a conjugacy class. Proposition 3 (Previtali 2001). If Conj^{2} holds, then Cov_{𝒩} (G) = Car(G).

Evidence for Conj^{2}

Theorem 4 (Dalla Volta, Lucchini, Tamburini 1998). Let G be a minimal counterexample to Conj^{2}, then

1. S ≤ G ≤ Aut(S), where S is a non-abelian simple group;

2. G = SH, for any H ∈ Car(G).

In particular G is almost-simple and G/S is nilpotent.

Carter of almost-simple groups

Using the Classification of Finite Simple Groups CFSG, we may pin down G. Let Syl_{2}(G) denote the Sylow 2-subgroup of G. Theorem 5 (Di Martino, Tamburini 1976). Let G be S_{n} or A_{n}, n ≥ 6, then H ∈ Car(G) iff H = N_{G}(T), where T ∈ Syl_{2}(G). Theorem 6 (Di Martino, Tamburini 1976, 1987; Vavilov 1979; Gagen 1980). Let SL_{n}(q) ≤ G ≤ GL_{n}(q), then H ∈ Car(G) only if H = N_{G}(T), where T ∈ Syl_{2}(G), or G = SL_{2}(3) and |H| = 6. Moreover N_{G}(T) is nilpotent iff one of the following holds:

1. q = 2;

2. q is odd and n > 2;

3. q is odd, n = 2 and P G = PGL_{2}(q);

4. q ≡_{8} ±1 and P G = PSL_{2}(q).

Theorem 7 (Di Martino, Tamburini, Zalesskii 1997). Let G be on of the following classical groups O_{n}(q), q odd, Sp_{n}(q) or U_{n}(q). If H ∈ Car(G), then one of the following holds:

1. G = Sp_{2}(3), |H| = 6;

2. G = U_{3}(4), |H| = 18;

3. H = N_{G}(T), where T ∈ Syl_{2}(G).

Since N(Syl_{2}(G)) is a conjugacy class, then Conj^{2} holds in the previous cases.

Towards a minimal counterexample Theorem 8 (Tamburini, Vdovin 2001). Let A be an almost-simple group, K = Soc(A), K̂ the inner-diagonal automorphism group of K. Then Conj^{2} holds for A in the following list:

• Alt, Spor, ^{2}B_{2}(2^{2n+1}), G_{2}(r^{t}), F_{4}(r^{t}), 2F_{4}(2^{2n+1}),^{3}D_{4}(r^{3t}) (t even if r = 3), E_{8}(r^{t})(r ≠ 3, 5);

• A/K a 2-group or A ≥ K̂:

A_{1}(r^{t}), B_{ℓ}(r^{t}), C_{ℓ}(r^{t}), D_{2ℓ}(r^{t}), 3D_{4}(r^{3t}), ^{2}D_{2ℓ}(r^{2t}) (t even if r = 3), E_{7}(r^{t})(r ≠ 3);

• A = K: B_{ℓ}(3^{t}), C_{ℓ}(3^{t}), D_{2ℓ}(3^{t}), ^{3}D_{4}(3^{3t}), 2D_{2ℓ}(3^{2t}), D_{2ℓ+1}(r^{t}), ^{2}D_{2ℓ+1}(r^{2t}), ^{2}G_{2}(3^{2n+1}), E_{6}(r^{t}), ^{2}E_{6}(r^{2t}), E_{7}(3^{t}), E_{8}(3^{t}), E_{8}(5^{t});

• K ≤ A ≤ K̂: A_{ℓ}(r^{t}), ^{2}A_{ℓ}(r^{2t}).

Our result improves on the last two theorems.

We say G is an intermediate classical group if S ≤ G ≤ I, where (S, I) = (SO_{n}(q), O_{n}(q)), q odd, (SU_{n}(q), U_{n}(q)) or (Sp_{n}(q), Sp_{n}(q)).

From now on G will denote an intermediate classical group Theorem 9 (Previtali, Tamburini, Vdovin 2002). Let H ∈ Car(G) then:

1. G = Sp_{2}(3) and |H| = 6;

2. G = U_{3}(4) and |H| = 18;

3. H = N_{G}(T), where T ∈ Syl_{2}(G).

Details

We say that J ∈ GL_{n}(q) is reflexive if J is

1. symmetric, J = J^{t};

2. alternating, J = A − A^{t}, for some A;

3. hermitian, J = J^{ρ}, ρ ∈ Aut(𝔽_{q}), |ρ| = 2. Any intermediate classical group has shape

{g ∈ GL_{n}(q) | gJg^{tτ} = J, det g^{d} = 1}, for some reflexive J and τ ∈ Aut(𝔽_{q}), |τ| ≤ 2. We denote such group with G_{n}^{d}(q, J, τ).

Identifying classical and norm groups

Given an involution σ of Mat_{n}(q), we say that

U_{n}^{σ}(q) = {g ∈ GL_{n}(q) | σ(X)X = I_{n}}

is the Norm group defined by σ. Theorem 10 (Wagner 1967). Any involution σ of Mat_{n}(q) has shape

X ↦ JX^{tτ} J^{−1},

where τ ∈ Aut(𝔽_{q}), |τ| ≤ 2, and J is a reflexive matrix. Moreover:

(i) U_{n}^{σ}(q) is a (full) classical group or

(ii) q is even, J = J^{t} ≠ A − A^{t}, for any A, and:

(a) n = 2m + 1 and U_{n}^{σ}(q) ≃ Sp_{2m}(q);

(b) n = 2m + 2 and U_{n}^{σ}(q) is a split extension N · Sp_{2m}(q), where N is an elementary abelian 2-group.

Decomposability

A left-artinian algebra A decomposes as ⨁_{1}^{k} A_{i}, A_{i} an indecomposable two-sided ideal. Then A_{i} = Ae_{i}A, e_{i} a minimal central idempotent. Lemma 2. Assume the natural G-module, equals M ⊕ N, M, N maximally isotropic. Then any Levi factor L of the parabolic subgroup P = Stab_{G}(M) does not contain a Carter subgroup of G.

Use the conjugacy of Carter subgroups for any SL_{m}(q) ≤ L ≤ GL_{m}(q), where m = dim M.

Theorem 11. Let H ∈ Car(G) and T ≤ H. Assume that:

• H = C_{H}(T)T;

• 𝔽_{q}T = ⨁_{1}^{k} Ae_{i}A, e_{i} minimal central idempotent, rank(e_{i}) = n_{i}.

• σ = σ_{J,τ} : X ↦ JX^{tτ} J^{−1}.

Then 𝔽_{q}H is decomposable, 𝔽_{q}H ≤ ⨁_{1}^{k} Mat_{n_{i}}(q) and:

(1) G = H = SO_{2}^{+}(q), q > 3 or;

(2) σ fixes each e_{i}, hence J ∈ ⨁_{1}^{k} Mat_{n_{i}}(q).

Remark that G = SO_{2}^{+}(q) is a genuine exception, since 𝔽_{q}G = 𝔽_{q} ⊕ 𝔽_{q} and Je_{1} = e_{2}J.

Given G intermediate classical, we say that the type of G, T ype(G), is orthogonal, symplectic or unitary according to the sesquilinear form preserved by G.

Lemma 3. Let H ∈ Car(G), G = G_{n}^{d}(q, J, τ). Assume

• 𝔽_{q}H ≤ ⨁_{1}^{k} Mat_{n_{i}}(q);

• J = diag(J_{1}, …, J_{k}) ∈ ⨁_{1}^{k} Mat_{n_{i}}(q);

• H_{i} be the projection of H into Mat_{n_{i}}(q);

• d_{i} := | det H_{i}|;

• G_{i} := G_{n_{i}}^{d_{i}}(q, J_{i}, τ). Then

(1) Type(G_{i}) =Type(G);

(2) H_{i} is a Carter subgroup of G_{i};

(3) H = (H_{1} × ⋯ × H_{k}) ∩ G. Theorem 12. Let G and H be minimal counterexamples to the Main Theorem. Then 𝔽_{q}H is indecomposable.

Nilpotent Normalizers

Theorem 13 (Glauberman 1970; Menegazzo, Tamburini 2001). Let S be simple non-abelian, 1 ≠ P ∈ Syl_{p}(P), p odd. Then

N_{S}(P) > P C_{S}(P).

In particular N_{S}(P) is not nilpotent. Lemma 4. Assume N_{G}(R) is nilpotent, where G = G_{n}^{d}(q), q = r^{f}, 1 ≠ R ∈ Syl_{r}(G). Then

1. q = 2;

2. q = 4 and G = SU_{2}(4), U_{2}(4) or SU_{3}(4);

3. q = 3 and G = SL_{2}(3) = Sp_{2}(3) = SU_{2}(3). Lemma 5. Let G = G_{n}^{d}(q), q = r^{f}, P ∈ Syl_{p}(G), with p ≠ 2, r. If 𝔽_{q}P is irreducible, then N_{G}(P) is not nilpotent.

In fact n = p^{f} and N_{G}(P) projects onto the non-nilpotent group N_{S}(T), S = Sym(n), T a Sylow p-subgroup of S.

Galois Automorphisms and Isometries

Lemma 6. Let T ≤ G = G_{n}^{d}(q) satisfy C := 𝔽_{q}T ≃ 𝔽_{q^{n}}. Set 𝒢 = Gal(𝔽_{q^{n}}/𝔽_{q}). Then

1. 𝒢 ≤ N_{G_{n}^{1}(q)}(C^{∗}) or

2. G = O_{2m}^{ε}(q), q odd, \frac{1}{2}𝒢 ≤ N_{G_{n}^{1}(q)}(C^{∗}).

In particular, N_{G}(T) > T whenever n > 1 and G ≠ SO_{2}^{−}(q).

By this embedding we mean that a Galois automorphism is realized via conjugation by a suitable isometry.

An effective version

Theorem 14 (Wedderburn 1905). Let A = 𝔽_{q}T be a simple subalgebra of M = Mat_{n}(q). Then C = C_{M}(A) is simple, there exist appropriate m, t, h such that n = mth and, up to conjugation in M,

A = Mat_{m}(ϕ(𝔽_{q^{h}}))⊗I_{t}, C = I_{m}⊗Mat_{t}(ϕ(𝔽_{q^{h}})),

where ϕ : 𝔽_{q^{h}} → Mat_{h}(q) is a regular representation. Lemma 7. Let G be any finite group, P × Q = H ∈ Car(G), where P ∈ Syl_{p}(H). If C_{G}(P) is nilpotent, then H = N_{G}(P). In particular P ∈ Syl_{p}(G).

Consequences

Corollary 1. Conj^{2} holds for every intermediate classical group and for every simple group S. If X is a minimal counterexample to Conj^{2}, then the order of every H ∈ Car(X) is divisible by a fixed prime p.

What about the other almost-simple groups?

Ubiquity of 2?

Lemma 8. Conj^{2} holds for every non-simple finite group G of adjoint type.

By this we mean G = A_{1}(2), A_{1}(3), B_{2}(2), G_{2}(2), 2A_{2}(2),^{2}B_{2}(2),^{2}G_{2}(3),^{2}F_{4}(2). In almost all cases Car(G) = N(Syl_{2}(G)).

If G is almost-simple and Car(G) ≠ ∅, is Car(G) = N(Syl_{2}(G))? Theorem 15 (Bardella 1985; Previtali 2001). Let G = PΣL_{2}(3^{3^{a}}), then Car(G) = Syl_{3}(G). Theorem 16 (Previtali 2001). Let F = 𝔽_{2^{f}} and G = PSL_{2}(F) ⋊ Φ, where ⟨ϕ⟩ = Φ ≤ Aut(F), |ϕ| = 2^{a}b, b odd. Set E = F ix_{F} (ϕ^{2^{a}}) and P_{E} ∈ Syl_{2}(PSL_{2}(E)). Then Car(G) = ∅ unless Φ = Aut(F) in which case Car(G) = (P_{E}Aut(F))^{G}.