Periodic Groups with One Finite Nontrivial Sylow 2-Subgroup

Abstract

The following results are proved. Let \(d\) be a natural number, and let \(G\) be a group of finite even exponent such that each of its finite subgroups is contained in a subgroup isomorphic to the direct product of \(m\) dihedral groups, where \(m\leq d\). Then \(G\) is finite (and isomorphic to the direct product of at most \(d\) dihedral groups). Next, suppose that \(G\) is a periodic group and \(p\) is an odd prime. If every finite subgroup of \(G\) is contained in a subgroup isomorphic to the direct product \(D_{1}\times D_{2}\), where \(D_{i}\) is a dihedral group of order \(2p^{r_{i}}\) with natural \(r_{i}\), \(i=1,2\), then \(G=M_{1}\times M_{2}\), where \(M_{i}=\langle H_{i},t\rangle\), \(t_{i}\) is an element of order \(2\), \(H_{i}\) is a locally cyclic \(p\)-group, and \(h^{t_{i}}=h^{-1}\) for every \(h\in H_{i}\), \(i=1,2\). Now, suppose that \(d\) is a natural number and \(G\) is a solvable periodic group such that every of its finite subgroups is contained in a subgroup isomorphic to the direct product of at most \(d\) dihedral groups. Then \(G\) is locally finite and is an extension of an abelian normal subgroup by an elementary abelian \(2\)-subgroup of order at most \(2^{2d}\).