On exact irreducible representations of locally normal groups

Irreducibly represented groups

“…The purpose of the present paper is to extend Gaschütz'result to infinite groups and unitary representations; for the particular case of finite groups, our arguments provide a new proof of the main result of [Gasch54] (at least for complex representations). For a generalisation of Gachütz' result of a rather different kind, see [Tushe93].…”

Infinite groups admitting a faithful irreducible representation

“…(b) As A is a maximal abelian normal subgroup of a soluble group G, it is easy to see that A is essential in G. By [T,Lemma 2], A has an irreducible representation ϕ over k such that Ker ϕ = C . Since C = Kerϕ contains no nontrivial normal subgroups of G, we see that A ∩ X is not contained in Ker ϕ for any nontrivial normal subgroup X of G, so the assertion follows from [T,Lemma 3].…”

“…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.…”

Groups having a faithful irreducible representation

“…When G itself is finite [T2,Theorem 1] or, alternatively, Theorems 1.1 and 1.2, yield a full criterion (no additional hypotheses are required) .…”