Non-periodic groups with the restrictions on the norm of cyclic subgroups of non-prime order

Abstract

One of the main directions in group theory is the study of the impact of characteristic subgroups on the structure of the whole group. Such characteristic subgroups include different $\Sigma$-norms of a group. A $\Sigma$-norm is the intersection of the normalizers of all subgroups of a system $\Sigma$. The authors study non-periodic groups with the restrictions on such a $\Sigma$-norm -- the norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order, which is the intersection of the normalizers of all cyclic subgroups of composite or infinite order of $G$. It was proved that if $G$ is a mixed non-periodic group, then its norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order is either Abelian (torsion or non-periodic) or non-periodic non-Abelian. Moreover, a non-periodic group $G$ has the non-Abelian norm $N_{G}(C_{\bar{p}})$
of cyclic subgroups of non-prime order if and only if $G$ is non-Abelian and every cyclic subgroup of non-prime order of a group $G$ is normal in it, and $G=N_{G}(C_{\bar{p}})$.
Additionally the relations between the norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups
of non-prime order and the norm $N_{G}(C_{\infty})$ of infinite cyclic subgroups, which is the intersection of the normalizers of all infinite cyclic subgroups, in non-periodic groups are studied. It was found that in a non-periodic group $G$ with the non-Abelian norm $N_{G}(C_{\infty})$ of infinite cyclic subgroups norms $N_{G}(C _{\infty})$ and $N_{G}(C _{\bar{p}})$ coincide if and only if $N_{G}(C _{\infty})$ contains all elements of composite order of a group $G$ and does not contain non-normal cyclic subgroups of order 4.
In this case $N_{G}(C_{\bar {p}})=N_{G}(C_{\infty})=G$.