On finite and locally finite subgroups of free Burnside groups of large even exponents
The Burnside Problem asks whether the free \(m\)-generator (Burnside) group \(B(m,n)\) of exponent \(n\) is finite. It was proved by P. S. Novikov and S. I. Adyan, 1968, that \(B(m,n)\) is infinite for any \(m\geq 2\) and any odd \(n\geq 665\). For sufficiently large even exponents \(n\) infinite groups \(B(m,n)\) were constructed only recently in the works of S. V. Ivanov, 1994, and I. G. Lysenok, 1996. The structure of \(B(m,n)\) for even \(n\) seems to be more complicated than for \(n\) odd. For example, for odd \(n\geq 665\) all finite or abelian subgroups of \(B(m,n)\) are cyclic. But this cannot be true for even \(n\), since every infinite \(2\)-group has infinite abelian subgroups by a theorem of D. Held and O. H. Kegel. Analyzing finite subgroups of \(B(m,n)\) for even \(n\) turned out to be an important part of proving the infiniteness of those groups. In an earlier article, the first author proved that if \(n=n_1n_2\geq 2^{48}\) with \(n_1\) odd and \(n_2=2^k\geq 2^9\), then any finite subgroup of \(B(m,n)\) is isomorphic to a subgroup of \(D(2n_1)\times D(2n_2)^l\), where \(D(2a)\) denotes the dihedral group of order \(2a\), and \(D(2n_2)^l\) is the \(l\)th direct power of \(D(2n_2)\). In the present article the authors prove, in particular, that all such groups do indeed appear as subgroups of \(B(m,n)\), where \(m\geq 2\) and \(n\) satisfies the above conditions. For \(m\) and \(n\) as above, the authors give a concise description of all infinite locally finite subgroups of \(B(m,n)\) (up to isomorphism). It is proved that any infinite locally finite subgroup \(S\) of \(B(m,n)\) is contained in a unique maximal infinite locally finite subgroup, while any finite \(2\)-subgroup of \(B(m,n)\) is contained in continuously many pairwise non-isomorphic maximal locally finite subgroups. In addition, \(S\) is locally conjugate to a maximal locally finite subgroup. Proofs are based on the analysis of the centralizers of subgroups of \(B(m,n)\). For example, it is proved that the centralizer of a finite 2-subgroup of \(B(m,n)\) contains a subgroup isomorphic to a free group \(B(\infty,n)\) of exponent \(n\) of countably infinite rank; the centralizer of a finite subgroup that is not a 2-group is always finite, as well as the centralizer of any subgroup that is not locally finite. The centralizer of \(S\) is infinite if and only if \(S\) is a locally finite \(2\)-group.
scientific article; zbMATH DE number 1076274
scientific article