In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. For each element a of a group G, conjugation by a is the operation φ a : G → G given by φ a (g) = aga (or a ga ; usage varies). One can easily check that conjugation by a is a group automorphism. The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma.