Description of the problem

The advanced theory of groups depends on the idea of an invariant (or normal or self-conjugate) subgroup. Beyond the elementary work on subgroups leading up to Lagrange's theorem, and homomorphisms, it is not possible to progress far with the study of group theory without introducing and using this concept. Its importance cannot be over-emphasised.

Although so important, the idea of an invariant subgroup is not easy to grasp for a student meeting it for the first time. It is easy to give a definition, but the significance of this and its far-reaching implications are difficult to appreciate before more experience is gained. There are in fact several approaches to the problem.

In this chapter we will develop the essential theory of invariant subgroups. In §6.2 and §6.3 we give what seems to be the most illuminating definition, and in §6.4 give an alternative and important approach. In §6.7 and §6.8 the important connection between invariant subgroups and homomorphisms is explained; in §6.9 the concept is linked with the idea of direct products.

In the remainder of the present section we try to explain some of the ideas that lie behind the definition, attempting to show intuitively some of the importance of the concept. The approach is informal, the rigorous treatment being left until the next sections.

The problem is basically that of ‘dividing’ one group by another: that is in ‘dividing’ a group G by a subgroup H to obtain another group, a ‘quotient group’.