# zbMATH — the first resource for mathematics

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