Groups whose subgroups are either abelian or pronormal

Locally Finite Simple Groups Whose Nonnilpotent Subgroups Are Pronormal

“…(i) Let X = (1, 2, 3), (1, 2)(4, 5) ; in particular, X Sym(3). Let g be the permutation (1, 4) (2,5,3,6). The subgroup…”

“…Now, PSL(4, 2) Alt( 8) is contained in PSL(5, 2) as a subgroup. Again by Lemma 2.5, the only possibility left is PSL (3,2), but this is isomorphic to PSL(2, 7) which has only pronormal or abelian subgroups (see [3,Proposition 4]).…”

“…Based on this work, finite simple groups whose nonpronormal subgroups are abelian have been classified in [3]. Actually, finite soluble groups whose nonabelian subgroups are pronormal began to be studied in [2,12] with the aim of expanding well-known structural theorems concerning metahamiltonian groups (that is, groups whose proper subgroups are either abelian or normal) to larger classes of groups (see also [8,10,11] for other generalisations of this type) and these results were applied to obtain the results in [3]. Here, we propose a different approach which does not rely on the complex study of the structure of finite soluble groups with only abelian or pronormal subgroups and makes it possible to characterise which finite simple groups have only pronormal or nilpotent subgroups.…”