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On disposition of involutions in a group. (English. Russian original) Zbl 0835.20051

Sib. Math. J. 34, No. 2, 385-394 (1993); translation from Sib. Mat. Zh. 34, No. 2, 208-219 (1993).
Let \(G\) be a group with involutions and \(i\) be an involution of \(G\) satisfying the following conditions: 1) all groups of the form \(\text{gr}(i,i^g)\), \(g\in G\), are finite; 2) Sylow 2-subgroups of \(G\) are cyclic or generalized quaternion groups; 3) \(C_G(i)\) is infinite and possesses finite periodic part; 4) the normalizer of any nontrivial \((i)\)-invariant locally finite subgroup of \(G\) either is included in \(C_G(i)\) or its periodic part is a Frobenius group with abelian kernel and finite noninvariant factor of even order; 5) \(C_G(i)\neq G\) and, for every element \(c\), in \(G\setminus C_G(i)\), strongly real with respect to \(i\), i.e., \(c^i=c^{-1}\), there exists an element \(s_c\) in \(C_G(i)\) such that the subgroup \(\text{gr}(c,c^{s_c})\) is infinite. We call a group \(G\) with involution \(i\) satisfying conditions 1-5 a \(T_0\)-group.
Main theorem. Let \(G\) be a group with involutions and let \(i\) be an involution of \(G\) meeting the following conditions: 1) all subgroups of the form \(\text{gr}(i,i^g)\), \(g\in G\), are finite; 2) the group \(C_G(i)\) possesses finite periodic part; 3) the normalizer of any nontrivial \((i)\)-invariant locally finite subgroup of \(G\) possesses periodic part. Then either the set of all elements of finite order generates a periodic almost nilpotent subgroup in \(G\) or \(G\) is a \(T_0\)-group.

MSC:

20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups
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