Let K be a subgroup of a group G. The set G/K of cosets of K forms a group, called a quotient group (or factor group), under the operation Let G be a group, let a be an elem...Let K be a subgroup of a group G. The set G/K of cosets of K forms a group, called a quotient group (or factor group), under the operation Let G be a group, let a be an element of G, and let Ca:G→G be given by Ca(g)=aga−1. The map Ca is called conjugation by the element a and the elements g,aga−1 are said to be conjugate to one another.
We now return to the notion of equipping G/H, when H⊴G, with a group structure. We have already saw that left coset multiplication on G/H is well-defined when H⊴G (Theorem 8.1.1); it turns out th...We now return to the notion of equipping G/H, when H⊴G, with a group structure. We have already saw that left coset multiplication on G/H is well-defined when H⊴G (Theorem 8.1.1); it turns out that given this, it is very easy to prove that G/H under this operation is a group.
