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Groups with many characteristically simple subgroups

1979; Cambridge University Press; Volume: 86; Issue: 2 Linguagem: Inglês

10.1017/s0305004100055997

1469-8064

John S. Wilson,

Finite Group Theory Research

1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G . It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G . A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.