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

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 NGA with the non-Hamiltonian Sylow p-subgroup (NGA)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∈ π(G), q≠ p) are cyclic, Sylow $2$-subgroups (p≠ 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 NGA of Abelian non-cyclic subgroups. Then G=Gₚ H, where Gₚ is an infinite HĀₚ-group of one of the types (1)--(4) of Proposition~2 in present paper, which coincides with the Sylow p-subgroup of the norm NGA, H is a finite group, all Abelian subgroups of which are cyclic, and $(|H|,p)=1$. Any element h∈ H of odd order that centralizes some Abelian non-cyclic subgroup M⊂ NGA is contained in the centralizer of the norm NGA. (Theorem 2).3) Let G be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm NGA of Abelian non-cyclic subgroups. ThenG=H K, where H is a finite group, all Abelian subgroups of which are cyclic,(|H|,2)=1, K is an infinite 2-group of one of the types (5)--(6) of Proposition~2 (in present paper). Moreover, the norm NKA of Abelian non-cyclic subgroups of the group K is finite, K∩ NGA=NKA and coincides with the Sylow 2-subgroup (NGA)₂ of the norm NGA of a group G.Moreover, any element h∈ H of the centralizer of some Abelian non-cyclic subgroup M ⊂ NGA is contained in the centralizer of the norm NGA. (Theorem 4).