Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups

Abstract

In the paper, the properties of infinite locally finite groups with non-Dedekind locally nil\-potent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic $p$-subgroups for a unique prime $p$. In particular, in the paper is prove the following assertions: 1) Let $G$ be an infinite locally finite group and contain the locally nilpotent norm $N_{G}^{A}$ with the non-Hamiltonian Sylow $p$-subgroup $(N_{G}^{A})_{p}$. Then $G$ is a finite extension of a quasicyclic $p$-subgroup, all Sylow $p'$-subgroups are finite and do not contain Abelian non-cyclic subgroups. In particular, Sylow $q$-subgroups ($q$ is an odd prime, $q\in \pi(G)$, $q\neq p$) are cyclic, Sylow $2$-subgroups ($p\neq 2$) are either cyclic or finite quaternion $2$-groups (Theorem 1).
 2) Let $G$ be a locally finite non-locally nilpotent group with the infinite locally nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then $G=G_{p} \leftthreetimes H,$ where $G_{p}$ is an infinite $\overline{HA}_{p}$-group of one of the types (1)--(4) of Proposition~2 in present paper, which coincides with the Sylow $p$-subgroup of the norm $N_{G}^{A}$, $H$ is a finite group, all Abelian subgroups of which are cyclic, and $(|H|,p)=1$. Any element $h\in H$ of odd order that centralizes some Abelian non-cyclic subgroup $M\subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 2).
3) Let $G$ be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then
$G=H\leftthreetimes K,$ where $H$ is a finite group, all Abelian subgroups of which are cyclic,
$\left(\left|H\right|,2\right)=1$, $K$ is an infinite 2-group of one of the types (5)--(6) of Proposition~2 (in present paper). Moreover, the norm $N_{K}^{A}$ of Abelian non-cyclic subgroups of the group $K$ is finite, $K\cap N_{G}^{A}=N_{K}^{A}$ and coincides with the Sylow 2-subgroup $(N_{G}^{A})_2$ of the norm $N_{G}^{A}$ of a group $G$.
Moreover, any element $h\in H$ of the centralizer of some Abelian non-cyclic subgroup $M \subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 4).