Definition 1

A set with an associative law of composition, possessing an identity element and under which every element is invertible, is called a group.

In other words, a group is a monoid (§ 2, no. 1, Definition 1) in which every element is invertible. A law of composition on a set which determines a group structure on it is called a group law. If G and H are two groups, a magma homomorphism of G into H is also called a group homomorphism. Such a homomorphism $f$ maps identity element to identity element; for, let $e$ (resp. $e'$) be the identity element of G (resp. H); writing the group laws of G and H multiplicatively, $e \cdot e = e$, whence $f(e) \cdot f(e) = f(e)$ and, multiplying by $f(e)^{-1}, f(e) = e'$. Hence $f$ is unital. It then follows from no. 3 of § 2 that $f(x^{-1}) = f(x)^{-1}$ for all $x \in G$.