Groups without Faithful Transitive Permutation Representations of Small Degree

✍ Scribed by László Babai; Albert J. Goodman; László Pyber

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 328 KB
• Volume: 195
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7042

✦ Synopsis

A subgroup H of a group G is core-free if H contains no non-trivial normal subgroup of G, or equivalently the transitive permutation representation of G on the cosets of H is faithful. We study the obstacles to a group having large core-free subgroups. We call a subgroup D a ''dedekind'' subgroup of G if all subgroups of D are normal in G. Our main result is the following: If a finite group G has no core-free subgroups of order greater than k, then G has two dedekind subgroups Ž . D and D such that every subgroup in G of order greater than f k has 1 2 Ž non-trivial intersection with either D or D where f is a fixed function indepen-1 2

. dent of G . Examples show that the dedekind subgroups need not have index bounded by a function of k, and the result would not be true with one dedekind