M ₉-free groups

Abstract

We give a characterisation of all finite groups whose subgroup lattice does not contain a sublattice isomorphic to

Finite simple 3′-groups are cyclic or Suzuki groups

“…We summarize: The definitions of σ and H imply that H ≤ N , and (6) shows that the non-trivial Sylow subgroups of H are not cyclic. In addition L ≤ K induces power automorphisms on H. From (10) we deduce that π ∩ π = ∅.…”

“…Then by definition there are q ∈ π and r ∈ π and subgroups Q ≤ L 1 and R ≤ L 2 such that the following hold: |Q| = q, |R| = r, 1 = [N,Q] ≤ O p (N ) =: P and 1 = [N,R] ≤ P . In particular (6) shows that K induces power automorphisms on P . It follows from Corollary 5.6 (c) and Lemma 5.4 that P = [P,K] = [P,Q] = [N,Q] = [N,R], and then (8) shows that P ∩ S = 1.…”

“…By (6), we see that R and Q act irreducibly on P , and (1) gives that Q is cyclic. Then ( 8) and (7) say that Φ(Q) induces non-trivial power automorphisms on P .…”

“…Then for every X ∈ {T,S} there is some x ∈ P such that X = (X ∩ P )N X (Q x ), by Lemma 7.8 (b). Using (6) we see that K induces power automorphisms on P . Thus P = [P,K] is elementary abelian by Corollary 5.6 (c), and then Lemma 7.9 is applicable.…”

“…Together with Andreeva and the first author he also characterised, in [1], all finite groups that are L 8 -free or M 8 -free. Finally, the finite M 9 -free groups have been classified by Pölzing and the second author in [6]. Furthermore, there is a general discussion of L 10 -free groups by Schmidt, which can be found in [9] and [10].…”