Existence and Uniqueness of O’Nan simple group

G. O. Michler and A. Previtali

Milano, 12 Ottobre 2003

Involutions

Given a finite group G let

Inv(G) = {u ∈ G | u^{2} = 1 ≠ u} and

Sr(G) = {uv | u, v ∈ Inv(G), u ≠ v} be the set of involutions and strongly real elements. Theorem 1 (Brauer-Fowler 1955). Let G be a finite group of even order. Set h = max{|C_{G}(x)| | x ∈ Inv(G) ∪ Sr(G)}. For any u ∈ Inv(G), |G| < h|C_{G}(u)|^{2}; if Inv(G) = u^{G}, then |G| < |C_{G}(u)|^{3}; if Inv(G) ≠ u^{G}, G simple, then

|G| < (|C_{G}(u)|^{2})!.

Simple Groups from Centralizers

Theorem 2 (Feit-Thompson 1963). Any group of odd order is soluble.

We say that G is of H-type if C_{G}(u) ≃ H for some u ∈ Inv(G). Corollary 1. There are only finitely many simple groups of H-type, for a given group H.

We say z ∈ Inv(G) is 2-central if z ∈ Z(P), P ∈ Syl_{2}(G).

Problem: Determine all simple groups G with prescribed C_{G}(z), for a 2-central z.

Normal 2-rank

We denote n_{2}(G) the normal 2-rank of G, the maximal rank of normal elementary abelian subgroups of any Sylow 2-subgroup of G.

We characterize groups of low normal 2-rank. We recall that ⟨a, b : a^{2^{n}} = b^{2} = 1, a^{b} = a^{2^{n−1}−1}⟩ is called semidihedral. Theorem 3. Let P be a 2-group such that n_{2}(P) ≤ 1, then S is

cyclic;

dihedral;

semidihedral;

generalized quaternion. Theorem 4 (Thompson). If G has cyclic Sylow 2-subgroup P, then G = P 𝕆(G) and G is not simple.

Simple Groups of low normal 2-rank

Theorem 5 (Gorenstein-Walter 1962). If G is a group with dihedral Sylow 2-subgroup P, then H = G/𝕆(G) satisfies:

PSL_{2}(q) ≤ H ≤ PΓL_{2}(q), q odd;

H = A_{7};

H is a Sylow 2-subgroup of G. Theorem 6 (Alperin-Brauer-Gorenstein 1970). If G is a simple group with semidihedral Sylow 2-subgroup P, then G is:

PSL_{3}(q), q ≡_{4} −1;

PSU_{3}(q), q ≡_{4} 1;

the Mathieu group M_{11}. Theorem 7 (Brauer-Suzuki 1959). If G is a group with generalized quaternion Sylow 2-subgroup P, then |Z(G/𝕆(G))| = 2 and G is not simple.

Sporadic Groups

Remark 1. For all sporadic simple groups G n_{2}(G) > 1, unless G = M_{11}. Namely any P ∈ Syl_{2}(G) has a non-cyclic elementary abelian normal subgroup A.

Michler’s Algorithm

Input: A finite group of even order H, given as finitely presented group.

Output: All simple groups G such that

C_{G}(z) ≃ H, for a 2-central element z ;

there exists A ⊴ P ∈ Syl_{2}(G), elementary abelian non-cyclic such that G := ⟨N_{G}(A), C_{G}(z)⟩;

there exists an \overline{𝔽_{r}} [G]-module M, M_{H} multiplicity-free, for some prime r < |H|^{2} − 1, p |̸|H||N_{G}(A)|.

Remark that G embeds in Alt_{|H|^{2}}.

Strongly embedded subgroups

We say U is strongly embedded in G if U < G has even order; |U ∩ U^{x}| is odd for any x ∈ G ∖ U. Concept introduced by Suzuki to classify CT-groups. Theorem 8 (Bender-Suzuki 1971). Let U be strongly embedded in G then a Sylow 2-subgroup P of G is cyclic or

generalized quaternion; or G possesses a normal series

G ⊳ L ⊳ M ≥ 1 where G/L and M = 𝕆(G) have odd order and L/M is P SL_{2}(q), PSU_{3}(q) or 2B_{2}(q) = Sz(q), q = 2^{m}.

Generation

Proposition 9. The following statements are equivalent:

U is strongly embedded in G;

U has even order and N_{G}(Q) ≤ U for any 2-subgroup Q ≠ 1 of U;

there exists P ∈ Syl_{2}(G) such that

N_{G}(P) ≤ U and C_{G}(t) ≤ U for t ∈ Inv(U). Proposition 10. Let G be a group of even order without strongly embedded subgroups. Let H be maximal among all centralizers of 2-central involutions of G. Let P ∈ Syl_{2}(G). Then

G = ⟨H, N_{G}(P), C_{G}(t) : t ∈ Inv(H)⟩.

Groups of ON-type

Let G be of ON-type, z a 2-central involution of G, H = C_{G}(z). Fix P ∈ Syl_{2}(H), then z ∈ Z(P) and H ≃ 4.PSL_{3}(4) : 2 |P | = 2^{9}; ∃!A elementary abelian normal subgroup of P

of rank 3; ∃!V elementary abelian normal subgroup of P

of rank 2; C_{P} (A) = C_{H}(A) ≃ 4^{3}, homocyclic group; C_{H}(V) = C_{P} (V) and Z(C_{P} (V)) = V; N_{H}(V) = P.

Existence

Set D = N_{H}(A) and η : D → AutA ≃ GL_{3}(2) the conjugation action. Set Δ = η(D). By Michler’s Algorithm we need to determine 𝔖(A) = {Φ ≤ GL_{3}(2)} such that:

Δ = C_{Φ}(z);

|Φ : Δ| is odd and > 1;

there exist a group extension

1 ⟶ C ⟶^{κ} E ⟶^{λ} Φ ⟶ 1

and an embedding μ : D ⟶ E such that κ ∘ id_{C} = μ_{↾C} and λμ = η.

Remark 2. E will play the rôle of N_{G}(A).

E-uniqueness

Since Δ ≃ Sym_{4} is maximal in GL_{3}(2), GL_{3}(2) is the UNIQUE element of 𝔖(A).

By MAGMA C = C_{H}(A) is not complemented in D. Thus E is a non-split extension of C by GL_{3}(2). By MAGMA we have Theorem 11. E is uniquely determined as a non-split extension of C ≃ 4^{3} and Φ ≃ GL_{3}(2). Moreover E = ⟨D, y⟩, for some y.

How do we get y? Theorem 12 (Glauberman 1966). Let G be any finite group, P ∈ Syl_{2}(G), Z^{∗}(G)/𝕆(G) = Z(G/𝕆(G)). Then x ∈ Z^{∗}(G) iff x^{G} ∩ C_{P} (x) = {x}.

By Glauberman’s Z^{∗}-Theorem there exists y ∈ E such that |y| = 3 and V = ⟨z, z^{y}⟩.

Generating G

By Bender-Suzuki’s Theorem we also get Proposition 13. If G is a simple group of ON-type then G =< C_{G}(z), N_{G}(A) >, where z is 2-central and A is normal elementary abelian of maximal rank in a Sylow 2-subgroup P of G.

From this we get that Inv(G) = z^{G}.

We say that x ∈ G is z-special if z ∈ ⟨x⟩ and call x^{G} a z-special class. Since x = su = us, where u is 2-singular and s 2-regular, we get a complete list for all G-classes of even order.

Local 3-structure

Using Theorem 14 (Brauer-Wielandt). Let Q be an odd order group admitting the action of a Klein 4-group V, then

|Q||C_{Q}(V)|^{2} = ∏_{t∈Inv(V)} |C_{Q}(t)|.

and the Gorenstein-Walter classification of groups with dihedral Sylow 2-subgroup we get Theorem 15. Let r ∈ H have order 3 and G of ON-type then C_{G}(r) = R × A_{6}, where R ∈ Syl_{3}(H).

Embedding J_{1}

Theorem 16 (Main). Let G be of ON-type, then H = C_{G}(z) = ⟨r, s, f⟩ of orders 3, 16, and 5; there exists a Glauberman element y ∈ G of

order 3 such that E = N_{G}(A) = ⟨r, s, y⟩ is the unique non-split extension of 4^{3} by PSL_{3}(2);

J = ⟨r, f, y⟩ is isomorphic th the smallest

Janko group; the defining relations ℛ(G) for G concides

with ℛ(H) ∪ ℛ(E) ∪ ℛ(J); G admits a faithful permutation representation

of degree 122.760 with stabilizer L isomorphic to PSL_{3}(7) : 2;

|G| = 2^{9} · 3^{4} · 5 · 7^{3} · 11 · 19 · 31. In particular there exists a unique group of ON-type.