turkmath.org

Given an infinite sequence of not necessarily distinct prime numbers ξ = (p1, p2, , . . . , . . .). Let κ be a given infinite cardinal number. By using the strictly diagonal embedding of finitary symmetric group F Sym(κni) into F Sym(κni+1) where ni = p1 . . . pi one may construct for any given cardinal κ uncountably many pairwise non-isomorphic simple locally finite groups of cardinality κ. Then the natural question is the following: What is the structure and the cardinality of the automorphism group of F Sym(κ)(ξ). We prove the following. Theorem 1. (G¨uven-Kuzucuo˘glu) The cardinality of the automorphism group of F Sym(κ)(ξ) is equal to $2^κ$. As a tool we use the level preserving automorphisms of F Sym(κ)(ξ). We will also mention corresponding spherically homogenous rooted tree and topological properties of such trees.