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Omissible extensions of \(SL_2(k)\) where \(k\) is a field of positive characteristic. (English) Zbl 1310.20036
In an earlier work the authors proved that a locally (soluble-by-finite) group \(G\) with all its proper subgroups soluble-by-(finite rank) is one of four types, the third of which being that \(G\) is soluble-by-\(\mathrm{PSL}(2,k)\) for \(k\) an infinite locally finite field with no infinite proper subfields. The object of the paper under review is to construct non-obvious groups of this type.
For example the authors prove the following (see Theorem 1.2). Let \(k\) be an infinite locally finite field with no infinite proper subfields and let \(d\) be a positive integer. Then there exists a countable locally finite group \(G\) with all its proper subgroups soluble-by-(finite rank) such that \(G\) has a normal subgroup \(H\) of finite exponent that is soluble of derived length \(d\) and such that \(G/H\) isomorphic to \(\mathrm{PSL}(2,k)\).
An important tool in this work is the Frattini-related notion of an omissible subgroup, by which the authors mean a normal subgroup \(N\) of a group \(G\) such that whenever \(X\) is a subgroup of \(G\) with \(XN=G\), then \(X=G\).
MSC:
20F19 Generalizations of solvable and nilpotent groups
20F50 Periodic groups; locally finite groups
20H20 Other matrix groups over fields
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
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