Semi-isomorphisms of certain infinite permutation groups

Abstract

Let X and Y be infinite cardinal numbers, S(X) the full symmetric group on a set of cardinal X, A (X) the alternating group of finite even permutations on the same set, and S(X, Y) the subgroup of S(X) of all permutations moving fewer than Y elements. A semi-automorphism of a group G is a permutation T of G such that (xyx)T= (xT)(yT)(xT) for all x, yEG. Semi-isomorphism is defined similarly. Dinkines [1] and Herstein and Ruchte [2] showed that any semi-automorphism of S(X, Y) or A (X) was either the restriction T of an inner automorphism of S(X), or was of the form T(-I) where x(-I) =x-1 for all x. Theorem 11.4.6 of [3] states that every automorphism of any group G such that A (X)CGCS(X) is the restriction of an inner automorphism of S(X). In the present paper, we prove the common generalization of these two theorems whose statement is obvious. (See the corollary at the end.)