A criterion of nonsimplicity of a group with a finite element

Infinite groups with finiteness conditions traditionally include periodic groups and locally finite groups. Later, in the Krasnoyarsk school on infinite groups, new finiteness conditions for a system of subgroups appeared: conjugately biprimitive finite groups, weakly conjugately biprimitive finite groups, biprimitive finite groups, weakly biprimitive finite groups, introduced by V.P. Shunkov, in which subgroups generated by pairs of elements (pairs of conjugate elements, pairs of elements of the same order, pairs of such elements in sections of the group by finite subgroups) were assumed to be finite. Infinite groups with finiteness conditions for a system of subgroups include groups with a finite element, introduced by A.I. Sozutov. An element a of a group G is called a finite element if groups of the form ⟨a,g-1ag⟩ g∈G , are finite. It is proved that the group G without involutions, not having a layer-finite periodic part, with M-finite element a of prime order, where M is the normalizer of a maximal layer-finite subgroup containing the periodic part of the group NG( ⟨a⟩), in the case when the normalizer of any finite non-trivial subgroup has an infinite layer-finite periodic part, has the form G = F ⋋ NG⟨a⟩ and F ⋋ ⟨a⟩ = ⟨aG⟩ is a Frobenius group with the kernel F and the complement ⟨a⟩.

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