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 N₆^A with the non-Hamiltonian Sylow p-subgroup (N₆^A) ₏. 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 (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 N₆^A of Abelian non-cyclic subgroups. Then G=G₏ H, where G₏ is an infinite HA₏-group of one of the types (1) -- (4) of Proposition~2 in present paper, which coincides with the Sylow p-subgroup of the norm N₆^A, 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 N₆^A is contained in the centralizer of the norm N₆^A. (Theorem 2). 3) Let G be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm N₆^A 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 N₊^A of Abelian non-cyclic subgroups of the group K is finite, K N₆^A=N₊^A and coincides with the Sylow 2-subgroup (N₆^A) ₂ of the norm N₆^A of a group G. Moreover, any element h H of the centralizer of some Abelian non-cyclic subgroup M N₆^A is contained in the centralizer of the norm N₆^A. (Theorem 4).