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-generator Burnside group B(m, n) of even exponent n, where m > 1 and n ≥ 2⁴⁸, n is divisible by 2⁹, are obtained: A clear complete description of all infinite groups that are embeddable in B(m, n) as (maximal) locally finite subgroups is given. Any infinite locally finite subgroup L of B(m, n) is contained in a unique maximal locally finite subgroup, while any finite 2-subgroup of B(m, n) is contained in continuously many pairwise nonisomorphic maximal locally finite subgroups. In addition, L is locally conjugate to a maximal locally finite subgroup of B(m, n). To 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 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 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.