Breadcrumbs Section. Click here to navigate to respective pages.

Chapter

Chapter

# Linear Groups Which Are Close to Irreducible

DOI link for Linear Groups Which Are Close to Irreducible

Linear Groups Which Are Close to Irreducible book

# Linear Groups Which Are Close to Irreducible

DOI link for Linear Groups Which Are Close to Irreducible

Linear Groups Which Are Close to Irreducible book

## ABSTRACT

In Chapter 3 we discuss groups which are close to irreducible in one sense or another. An infinite dimensional linear group G whose proper G-invariant subspaces are finite dimensional and whose union is the underlying space is called quasi-irreducible. Among other things it is shown that locally radical quasi-irreducible subgroups of GL(F, A) are abelian-by-finite and Tor(G) has finite special rank.. We then consider the situation when the space A has a finite dimensional G-invariant sub-space B such that A/B is irreducible. In the second part of the chapter almost irreducible groups are discussed. This is the dual situation to the quasi-irreducible case, so every non-zero G-invariant subspace has finite codimension and the intersection of the non-zero G-invariant subspaces is trivial. The monolithic case, when the intersection of all the non-zero G-invariant subspaces has finite codimension is also considered. Finally in this chapter the notions of G-contrainvariant subspace and G-core-free subspace are introduced. A subspace B is G-contrainvariant if the G-invariant subspace GB generated by B is the whole of A and the situation when every subspace is either G-contrainvariant or G-invariant is discussed. Likewise B is G-core-free if the largest G-invariant subspace of G contained in B is 0 and again groups in which every subspace either is G-core-free or G-invariant are discussed.