This can be restated in terms of elements and internal conditions: If is a normal, complete subgroup of a group, , then is a direct product. The proof follows directly from the definition: is centerless giving is trivial. If is an element of then it induces an automorphism of by conjugation, but and this conjugation must be equal to conjugation by some element of . Then conjugation by is the identity on and so is in and every element, , of is a product in .