On infinite groups with given properties of the norm of infinite subgroups (Q2754823)

Let \(G\) be an infinite group. A subgroup of \(G\) which normalizes every infinite subgroup of \(G\) and is maximal with this property is called the norm of the infinite subgroups of \(G\) and is denoted by \(N_G(\infty)\). It is proved that the norm \(N_G(\infty)\) is Abelian for non-periodic groups \(G\) and \(N_G(\infty)=Z(G)\) if \(N_G(\infty)\) is itself non-periodic. As a consequence the following result is obtained: the norm \(N_G(\infty)\) of a non-Abelian non-periodic group \(G\) has finite index in \(G\) if and only if \(G\) is mixed and finite over its center. Some results are also obtained for locally finite groups with some restrictions on \(N_G(\infty)\). In particular, infinite locally finite groups \(G\) with finite center such that \(|G:N_G(\infty)|<\infty\) are described.