TY - JOUR

T1 - On finite and locally finite subgroups of free burnside groups of large even exponents

AU - Ivanov, S. V.

AU - Ol'shanskii, A. Yu

N1 - Funding Information:
The first author was supported in part by an Alfred P. Sloan Research Fellowship, a Beckman Fellowship, and NSF grant DMS-95-01056. The second author was supported in part by Russian Fund of Fundamental Research grant 96-01-0420.

PY - 1997/9/1

Y1 - 1997/9/1

N2 - 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.

AB - 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.

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U2 - 10.1006/jabr.1996.6941

DO - 10.1006/jabr.1996.6941

M3 - Article

AN - SCOPUS:0031236534

SN - 0021-8693

VL - 195

SP - 241

EP - 284

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -