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

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