Large Subgroups of Infinite Symmetric Groups

Abstract

This is a survey of work by many people on various notions of largeness for subgroups of infinite symmetric groups. The primary concern is with maximal subgroups of infinite symmetric groups, of which several new examples are given. Subgroups of small index in symmetric groups (and in other permutation groups) are considered, as are questions about covering symmetric groups with families of subgroups. Examples are given of maximal subgroups of other large permutation groups, such as GL(κ, F) (where κ is an infinite cardinal) and Aut(Q,≤ ). The paper concludes with a discussion of other notions of largeness in symmetric groups, such as oligomorphic, Jordan, maximal closed, and various strong transitivity conditions on infinite subsets.