ProofWiki defines a group isomorphism as a bijective homomorphism. In Topics in Algebra 2$\varepsilon$, Herstein defines a group isomorphism as an injective homomorphism:
Definition. A homomorphism $\phi$ from $G$ into $\bar{G}$ is said to be an isomorphism if $\phi$ is one-to-one.
Herstein does not define a homomorphism to be surjective, so the definition on ProofWiki and the definition in Topics in Algebra 2$\varepsilon$ contradict. Which definition is standard?
Without a doubt the standard definitions are: a group homomorphism is a function $f:G\to H$ between groups, such that $f(g_1g_2)=f(g_1)f(g_2)$.
If moreover $f$ is a injective then the homomorphism is also called a monomorphism.
If $f$ is a surjective homomorphism then it is called an epimorphism.
If $f$ is bijective, then it is called an isomorphism.
It should be noted that these are not arbitrary notions and there is really very little room to deviate from the standard. The notions of isomorphism, monomorphism, and epimorphism are categorical notions. Isomorphism means being invertible, monomorphism means being left invertible, and epimorphism means being right invertible. It is very easy then to show that for groups, a group homomorphism is invertible iff it is bijective, and it is left invertible iff and it injective. It is slightly more challenging to show that it is right invertible iff it if surjective.
And perhaps it could be mentioned that from a universal algebra perspective a group has two other operations besides the binary one, that is the unary operation of inversion and the nullary operation $e$ giving the neutral element. A homomorphism should be defined as preserving the three operations. However we know well that if the binary operation is preserved, so are the other two. A well known example where a little extra care is necessary is when dealing with homomorphisms of rings with identity, where we have to require explicitly that the nullary operation identity is preserved.
