On Finite and Locally Finite Subgroups of Free Burnside Groups of Large Even Exponents

The following basic results on infinite locally finite subgroups of a free m-generŽ . 48 ator Burnside group B m, n of even exponent n, where m ) 1 and n G 2 , n is divisible by 29, are obtained: A clear complete description of all infinite groups that Ž . Ž . are embeddable in B m, n as maximal locally finite subgroups is given. Any Ž . infinite locally finite subgroup L of B m, n is contained in a unique maximal Ž . locally finite subgroup, while any finite 2-subgroup of B m, n is contained in continuously many pairwise nonisomorphic maximal locally finite subgroups. In Ž . addition, L is locally conjugate to a maximal locally finite subgroup of B m, n . To Ž . prove these and other results, centralizers of subgroups in B m, n are investigated. For example, it is proven that the centralizer of a finite 2-subgroup of Ž . Ž . B m, n contains a subgroup isomorphic to a free Burnside group B `, n of countably infinite rank and exponent n; the centralizer of a finite non-2-subgroup Ž . Ž . of B m, n or the centralizer of a nonlocally finite subgroup of B m, n is always finite; the centralizer of a subgroup S is infinite if and only if S is a locally finite 2-group. Q 1997 Academic Press