## 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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### References:

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