A characterisation of almost simple groups with socle ²E₆(2) or M(22)

Abstract

We show that the sporadic simple group M(22), the exceptional group of Lie type 2 E 6 (2) and their automorphism groups are uniquely determined by the approximate structure of the centralizer of an element of order 3 together with some information about the fusion of this element in the group.

An identification theorem for the sporadic simple groups F2 and M(23)

“…Since Z is not weakly closed in C C S (π) (ρ) with respect to C K (ρ), we have that Z is not weakly closed in C S (ρ) with respect to C G (ρ). Hence C G (ρ)/ ρ is isomorphic to either M(22) or Aut(M(22)) by [14,Theorem 2]. Since C H (ρ) has order 2 8 · 3 10 , we have that C G (ρ)/ ρ ∼ = Aut(M(22)).…”

“…Proof. As remarked after the main theorems in [14], as a consequence of [14, Theorem 1], all the lemmas of that paper provide statements about the groups with socle 2 E 6 (2). In particular, (i) holds, part (ii) comes from [14,…”

“…This work is one of a series of papers where this type of problem is addressed. Particularly relevant to this article are [13,14,15,16]. We refer the reader to [15] where we motivate this study by describing certain configurations which appear when classifying the finite simple groups.…”

“…Note that by Lemma 4.6, C H (π) is similar to a 3-centralizer in M(22) when H is similar to a 3-centralizer of type M(23) and C H (π) is similar to a 3-centralizer of type 2 E 6 (2) when H is similar to a 3-centralizer of type F 2 . Therefore our goal in this section will almost be reached once we show that Z is not weakly closed in C S (π) with respect to C G (π), for then we will apply the main theorems of [14] to K. Lemma 5.3. We have C S (π) ∈ Syl 3 (K).…”

“…Then Lemma 4.6 (ii) says that C H (π)/ π is similar to a 3-centralizer in M(22). Furthermore, by Lemma 5.5, Z is not weakly closed in C S (π) with respect to K. Therefore [14,Theorem 1] implies that F * ( K) is isomorphic to M(22). Since, by Lemma 4.6 (ii), |C H (π)| = 2 8 · 3 9 , we have K ∼ = M(22).…”