The Homomorphism Theorems

Abstract

There exist linear groups with homomorphic images having no faithful representations of finite degree over any field whatever. For example by 2.2 every free abelian group, but not every abelian group, has faithful representations of finite degree over some field. This raises two questions. Firstly, for which classes-of-groups 𝔛 are homomorphic images of linear 𝔛-groups necessarily isomorphic to linear groups? Secondly, given an arbitrary linear group G for which normal subgroups N of G, is G/N isomorphic to a linear group? In some ways surprisingly the second question is fruitful while the first remains unproductive, except of counterexamples.