Groups having a faithful irreducible representation

Abstract

Abstract. We address the problem of finding necessary and sufficient conditions for an arbitrary group, not necessarily finite, to admit a faithful irreducible representation over an arbitrary field.

Infinite groups admitting a faithful irreducible representation

“…Suppose, conversely, that this condition holds. Then G has a faithful irreducible G-module over k by [Sz,Theorem 1.1].…”

“…Since D + and M are locally finite p-groups, D + ⋊ D * and M ⋊ Aut(M ) also exhibit the failure of Theorems 2.1, 1.4, 1.7, 3.1, 3.2 and 3.3 when the required finiteness conditions are removed (except that M ⋊Aut(M ) is unrelated to Theorem 1.7 as M is not abelian). Moreover, Example 3.7 shows that [T,Theorem 1] and [Sz,Theorem 1.2] cease to be valid if their respective finiteness conditions are removed. Indeed, in Example 3.7 the given group G has a minimal normal subgroup A that is an abelian p-group but nevertheless G has a faithful irreducible module over a field of characteristic p (in this regard, note that the socle of C p ≀ C ∞ is trivial).…”

“…Partial solutions appear in [T] and [Sz]. It should be noted that [T,Theorem 1] as well as [Sz,Theorem 1.2] require finiteness conditions. In this paper we obtain the following general criterion, where all finiteness conditions have been removed.…”

“…For sufficiency in both cases we rely on [Sz,Theorem 1.1]. We are more concerned here with the necessity of the stated conditions.…”

Representations of McLain groups

“…In 1954 McLain [M] constructed a family of groups that has been a rich source of examples in group theory ever since (see [M2], [R], [HH], [Ro], [W], [DG], [CS], [Sz2], for instance). A general McLain group M = M (Λ, ≤, R) depends on a set Λ, partially ordered by ≤, and an arbitrary ring R with 1 = 0.…”