Groups Acting on Groups

Abstract

The action of a group A on a set G is described by a homomorphism

$$ \pi :A \to {S_{G}} $$

; see Section 3.1. Suppose that G is not only a set but also a group. Then Aut G ≤ S _G , and we say that π describes the action of A on the group G if Im π is a subgroup of Aut G. In other words, in this case the action of A on G not only satisfies O ₁ and O ₂ but also

$$ {\left( {gh} \right)^{a}} = {g^{a}}{h^{a}}for\;all\;g,h \in Ganda \in A $$

.