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About periodic Shunkov group saturated with finite simple groups of Lie type rank 1. (Russian. English summary) Zbl 1368.20044

Summary: The property of a group \(G\) to be saturated with given set of groups \(X\) is a natural generalization of the locally-cover definition (in the class of locally finite groups) on periodic groups. A locally-finite group, wich has a locally-cover containing finite simple Lie-type groups of finite rank, is a Lie-type group on some locally finite field. We call a group “2Shunkov group” if every pair of conjugate elements generates a finite subgroup, and this property is saved after crossing on factor groups by finite subgroups. A group \(G\) is saturated with groups from the set \(X,\) if every finite subgroup \(K\) from \(G\) is contained in some subgroup \(G\) isomorphic to some group from \(X.\) In our work we solved the problem of building periodic Shunkov groups saturated with finite simple Lie groups of rank 1. Let \(\mathfrak{M}\) – be a set contains from finite simple groups Suzuki, Re, Unitary, Linear of Lie type rank 1. We proved that a periodic Shunkov group saturated with groups from set \(\mathfrak{M}\) is isomorphic to a simple group of Lie-type rank 1 for some locally finite field \(Q\). Also we got a description of a Sylow 2-subgroup of a periodic group saturated with groups from \(\mathfrak{M}\), what is a necessary step in establishing the structure of an arbitrary periodic group with given saturation set.

MSC:

20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
20D06 Simple groups: alternating groups and groups of Lie type
20E07 Subgroup theorems; subgroup growth
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References:

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