The Local Structure Theorem for Finite Groups with a Large 𝑝-Subgroup

The Local Structure Theorem: The wreath product case

“…If Q is a large subgroup of G, then it is easy to see that O p (N G (Q)) is also a large p-subgroup of G. Thus we also assume that (iii) Q = O p (N G (Q)). One of the consequences of G having a large p-subgroup is that G has parabolic characteristic p. In fact any p-local subgroup of G containing Q is 1 of characteristic p [MSS2,Lemma 1.5.5 (e)]. Further, if Q ≤ S ∈ Syl p (G), then Q is weakly closed in S with respect to G (Q is the unique G-conjugate of Q in S) [MSS2,Lemma 1.5.2 (e)].…”

“…One of the consequences of G having a large p-subgroup is that G has parabolic characteristic p. In fact any p-local subgroup of G containing Q is 1 of characteristic p [MSS2,Lemma 1.5.5 (e)]. Further, if Q ≤ S ∈ Syl p (G), then Q is weakly closed in S with respect to G (Q is the unique G-conjugate of Q in S) [MSS2,Lemma 1.5.2 (e)]. A significant part of the programme described in [MSS1] aims to determine the groups which possess a large psubgroup.…”

“…(2) and Y L is the tensor product module. This is an example in the tensor product case of [MSS2,Theorem A (6)]. We declare L to be in the unambiguous wreath product case if these two ambiguous configurations do not occur.…”

An improved 3-local characterization of McL and its automorphism group

“…For odd primes p, the extraspecial groups of exponent p and order p 2n+1 are denoted by p 1+2n + . The quaternion group of order 8 is Q 8 and Mat (10) is the Mathieu group of degree 10. A central product of groups H and K will be denoted H •K.…”

“…, C 14 as labeled in Table 1. We let x = (1, 2, 3) ∈ C 4 , y = (1, 2, 3)(4, 5, 6) ∈ C 6 , z = (1, 2, 3)(4, 5, 6) (7,8,9)…”