Existence and Uniqueness of Higman-Sims simple group

G. O. Michler and A. Previtali

Essen, 16 Giugno 2003

Involutions

Given a finite group G let

I(G) = {u ∈ G | u^{2} = 1 ≠ u}

and

Sr(G) = {uv | u, v ∈ I(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 ∈ I(G) ∪ Sr(G)}. For any u ∈ I(G), |G| < h|C_{G}(u)|^{2}; if I(G) = u^{G}, then |G| < |C_{G}(u)|^{3}; if I(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 ∈ I(G). Corollary 1. There are only finitely many simple groups of H-type, for a given group H.

We say z ∈ I(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.

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}}.

Sporadic Groups

Remark 1. All sporadic simple groups but 𝕄_{11} satisfy the previous conditions.

The smallest Mathieu group is classified among all simple groups of small 2-rank by Alperin, Brauer, Gorenstein, and Walter.

Fact: A 2-central involution z ∈ HS satisfies

C_{HS}(z) ≃ (4 ∗ 2^{1+4})Sym_{5}

a non-split extension of a central product of a cyclic group of order 4 and an extraspecial group of order 32 by the symmetric group on 5 letters.

A presentation for H

H = ⟨r, s, c | ℛ(H)⟩, where ℛ(H):

r^{3} = 1;

s^{8} = 1;

c^{5} = 1;

(sr)^{4} = 1;

(srcr^{−1})^{2} = 1;

rcr = sc^{−2}rs;

(s^{2}c^{−1}r)^{2} = 1;

(r^{−1}cs^{2})^{2} = 1;

scrcsr = c^{2};

s^{2}cr = rcsrs.

2-local structure

Fix P ∈ Syl_{2}(H), then z ∈ Z(P) and |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 complemented in D. Theorem 3 (Gaschütz 1952). Let C be an abelian normal subgroup of a group E and D a subgroup of E such that |C| ⊥ |E : D|, then:

C is complemented in D iff C is complemented in E;

if Φ_{1}, Φ_{2} are complements of C in E such that Φ_{1} ∩ D = Φ_{2} ∩ D, then Φ_{2} = Φ_{1}^{e}, for some e ∈ E.

In our setting |C| = 64 and |E : D| = 7, hence all complements to C in E = C : Φ are C-conjugate.

A Presentation for E

Theorem 4. E is uniquely determined as split extension of C ≃ 4^{3} and Φ ≃ GL_{3}(2). Moreover E = ⟨r, s, y | ℛ(E)⟩, where ℛ(E): r^{3} = 1; s^{8} = 1; y^{3} = 1; (ys)^{2} = 1; (sr)^{4} = 1; (yr)^{4} = 1; [s^{4}, r] = 1; ry^{−1}s = yrs^{2}yr; sy^{−1}s^{−2}r^{−1}s^{2}ry^{−1} = 1.

Fusion of involutions

How do we get y? Theorem 5 (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}.

Applying Glauberman’s Z^{∗}-Theorem to our candidate simple group G = ⟨H, E⟩ we get y ∈ E such that |y| = 3 and V = ⟨z, z^{y}⟩.

Compatible Pairs

Data: Two finitely presented group H and E with isomorphic subgroup H ≥ D ≃ D_{E} ≤ E.

Denote mfchar(X) the set of multiplicity-free, faithful, complex characters of the group X.

We say that (η, ϵ) is a compatible pair for (H, E) if

η ∈ mfchar(H) and ϵ ∈ mfchar(E);

η_{↾D} = ϵ_{↾D_{E}}. Proposition 6. ∃!(η, ε) compatible pair of (H, E) of minimal degree η(1) = 22 = ε(1).

(η, ε) = (η_{4} + η_{12} + η_{15}, ε_{1} + ε_{6} + ε_{10})

with common restriction to D ≃ D_{E} ≃ 4^{3} : S_{4}

η_{↾D} = δ_{1} + δ_{5} + δ_{11} + δ_{14} + δ_{22}

Brauer characters

Let X ∈ {H, E}.

Choose a Core-free Y ≤ X and a prime r |̸|H||E| (e.g. r = 11);

Apply Meataxe to (1_{Y})^{X} and to tensor powers of its constituents;

Lift and compare these r-modular representations with complex constituents of the previous compatible pair;

If we find a faithful constituents ϕ we may apply: Theorem 7 (Burnside 1911). For any Irr(X) ∋ χ, χ ≤ ϕ^{i}, where i ≤ |{ϕ(g) | g ∈ X}|.

Once we obtain 𝔜(H) and 𝔈(E) as subgroups of GL_{22}(r), determine T ∈ GL_{22}(r) such that D = D_{E}^{T}. Set G = ⟨𝔜(r), 𝔜(s), 𝔜(c), 𝔈(y)^{T} ⟩.

This is our candidate simple group.

G-Uniqueness

Identify r, s in H, E and let D = ⟨r, s⟩ = H ∩ E. Then G is an epimorphic image of the amalgamated free product H ∗_{D} E, where D = H ∩ E.

(My goodness, this might be infinite!)

We say that the amalgamated free product H_{1} ∗_{D_{1}} E_{1} has type (H, E, D) if H ≃ H_{1}, E ≃ E_{1}, and D ≃ D_{1}.

Goldschmidt index

Theorem 8 (Goldschmidt 1980). The number of non-isomorphic amalgamated free products of type (H, E, D) equals the number of (H^{∗}, E^{∗})-double cosets in AutD, where H^{∗} = {γ ∈ Aut(H) | D^{γ} = D} and E^{∗} = {γ ∈ Aut(E) | D^{γ} = D}.

We call such common value the Goldschmidt index of the type (H, E, D). Proposition 9. The Goldschmidt index of our given type is 1, hence H ∗_{D} E is UNIQUELY determined.

Thompson’s Theorem

Denote Char(X) the set of complex characters of X. Let A = H ∗_{D} E, η ∈ Char(H), ϵ ∈ Char(E), D = H ∩ E. Then η_{↾D} = ϵ_{↾D} iff ∃α ∈ Char(A) such that α_{↾H} = η and α_{↾E} = ϵ. Theorem 10 (Thompson 1980). Assume η_{↾D} = ϵ_{↾D} and set n = η(1), C_{H}, C_{E}, C_{D} the centralizers of the images of H, E, D in GL_{n}(ℂ). Then the number of non-equivalent α ∈ Char(A) such that α_{H} = η and α_{E} = ϵ equals the number of (C_{H}, C_{E})-double cosets in C_{D}.

Double Cosets

Proposition 11. Let F be a splitting field for H, E of characteristic r |̸|H||E| and order q. Assume there exist multiplicity-free faithful F -modules V, W for H, E of dimension n such that V_{D} = W_{D}, D = H ∩ E. Let k, m, l denote the number of irreducible constituents of V_{D}, V, and W. Then the number of (C_{H}, C_{E})-double cosets in C_{D} equals (q − 1)^{k+1−m−l}. Corollary 2. In our setting (r, n, k, l, m) = (11, 22, 5, 3, 3), hence there exists a UNIQUE representation of H ∗_{D} E to GL_{22}(11) restricting to the unique compatible pair found above.

Dixon-Schneider

Applying Dixon-Schneider’s Algorithm we obtain G and HS have the same character table.

Let k = |Irr(X)| = |CC(X)|, χ_{j}^{i} = χ^{i}(g_{j}) and g_{i′} = g_{i}^{−1}, c_{rst} the structure constants of X, M_{r} = (c_{rst}). Then (χ_{1}^{i}, …, χ_{k}^{i}) is a right eigenvector for any M_{r}. Example 1. Let X = Sym_{3}, M_{1} = I_{3},

M_{2} = ( \begin{matrix}0 & 1 & 0 \\ 3 & 0 & 3 \\ 0 & 2 & 0 \end{matrix} ), and M_{3} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 2 & 0 \\ 2 & 0 & 1 \end{matrix} ).

Left Eigenvectors (1, 1, 1), (1, −1, 1), and (2, 0, 1).