On groups with finite involution and locally finite 2-isolated subgroup of even period (Q5961083)

A proper subgroup \(H\) of a group \(G\) is `strongly isolated' if for every nonidentity element \(g\in G\) the condition \(C_G(g)\cap H\neq 1\) implies \(C_G(g)\leq H\) and `\(2\)-isolated' if this condition holds for all \(C_G(g)\) which contain an involution. In the sixties finite groups with strongly isolated subgroups have been studied by W. Feit, M. Suzuki, V. M. Busarkin and M. Herzog. \textit{M. Suzuki} [Osaka Math. J. 15, 143-150 (1963; Zbl 0122.27502)] and \textit{V. M. Busarkin} [Algebra Logika 4, No. 2, 33-50 (1965; Zbl 0145.02904)] showed that a finite group containing a strongly isolated subgroup of even order is either a Frobenius group or a ZT-group. \textit{V. M. Busarkin} [Mat. Zametki 3, 497-501 (1968; Zbl 0174.31003)] introduced the notion of a \(2\)-isolated subgroup and showed that a finite group which has a \(2\)-isolated subgroup \(H\) of even order such that there are involutions outside \(H\) has a similiar structure as a group with strongly isolated subgroup of even order. The object of the present paper is to prove an infinite analog of the theorem of Busarkin. The author needs the two further assumptions that the \(2\)-isolated subgroup \(H\) is locally finite and that \(G\) contains a finite involution \(i\), i.e. that \(i\) and \(i^g\) generate a finite group for every \(g\in G\).