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Arrangement of elements of prime orders in groups. (English. Russian original) Zbl 0722.20028

Algebra Logic 28, No. 4, 309-325 (1989); translation from Algebra Logika 28, No. 4, 463-483 (1989).
The following theorems are proved. Theorem 1. Let \({\mathcal G}\) be a group and a be an element of prime order \(p\neq 2\) of it such that the subgroups of the form \(gr(a,a^ g)\), \(g\in {\mathcal G}\), are finite and almost all are solvable. Then at least one of the following statements is valid: 1) The periodic part of G is finite; 2) There exists a finite subgroup of \({\mathcal G}\) containing a whose normalizer contains infinitely many elements of finite order; 3) The group \({\mathcal G}\) has an infinite periodic a-invariant abelian subgroup.
Theorem 2. A group \({\mathcal G}\), containing an element a of prime order \(p\neq 2\), has finite periodic part if and only if the following conditions are satisfied: 1) a is a point of \({\mathcal G}\); 2) Subgroups of the form \(gr(a,a^ g)\), \(g\in G\) are finite and almost all are solvable. Here an element \(g\in G\) is called a point of \({\mathcal G}\) if \(| g| <\infty\) and for an arbitrary nontrivial (g)-invariant finite subgroup \({\mathcal K}<{\mathcal G}\) the set of finite subgroups of \({\mathcal N}_{{\mathcal G}}({\mathcal K})\) containing g is finite; if \(g=1\) then in addition the set of elements of finite order of \({\mathcal G}\) is finite.

MSC:

20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
20E34 General structure theorems for groups
20F05 Generators, relations, and presentations of groups
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