On disposition of involutions in a group
Let \(G\) be a group with involutions and \(i\) be an involution of \(G\) satisfying the following conditions: 1) all groups of the form \(\text{gr}(i,i^g)\), \(g\in G\), are finite; 2) Sylow 2-subgroups of \(G\) are cyclic or generalized quaternion groups; 3) \(C_G(i)\) is infinite and possesses finite periodic part; 4) the normalizer of any nontrivial \((i)\)-invariant locally finite subgroup of \(G\) either is included in \(C_G(i)\) or its periodic part is a Frobenius group with abelian kernel and finite noninvariant factor of even order; 5) \(C_G(i)\neq G\) and, for every element \(c\), in \(G\setminus C_G(i)\), strongly real with respect to \(i\), i.e., \(c^i=c^{-1}\), there exists an element \(s_c\) in \(C_G(i)\) such that the subgroup \(\text{gr}(c,c^{s_c})\) is infinite. We call a group \(G\) with involution \(i\) satisfying conditions 1-5 a \(T_0\)-group. Main theorem. Let \(G\) be a group with involutions and let \(i\) be an involution of \(G\) meeting the following conditions: 1) all subgroups of the form \(\text{gr}(i,i^g)\), \(g\in G\), are finite; 2) the group \(C_G(i)\) possesses finite periodic part; 3) the normalizer of any nontrivial \((i)\)-invariant locally finite subgroup of \(G\) possesses periodic part. Then either the set of all elements of finite order generates a periodic almost nilpotent subgroup in \(G\) or \(G\) is a \(T_0\)-group.
scientific article; zbMATH DE number 825571
scientific article