On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period

Abstract

A proper subgroup H of a group G is said to be strongly isolated if it contains the centralizer of any nonidentity element of H and 2-isolated if the conditions >C _G(g) ∩ H ≠ 1 and 2∈π(C_G(g)) imply that C_G(g)≤H. An involution i in a group G is said to be finite if |ii ^g| < ∞ (for any g∈ G). In the paper we study a group G with finite involution i and with a 2-isolated locally finite subgroup H containing an involution. It is proved that at least one of the following assertions holds:

1) all 2-elements of the group G belong to H;

2) (G,H) is a Frobenius pair, H coincides with the centralizer of the only involution in H, and all involutions in G are conjugate;

3) G=FFC_G(i) is a locally finite Frobenius group with Abelian kernel F;

4) H=V ⋋ D is a Frobenius group with locally cyclic noninvariant factor D and a strongly isolated kernel V, U=O₂(V) is a Sylow 2-subgroup of the group G, and G is a Z-group of permutations of the set Ω=U ^g ∈ g ∈ G.