Surjectivity (onto) for isomorphism definition

Question

I have two books on algebra where one makes the definition of isomorphism to be a bijective homomorphism, while the other makes it as an injective homomorphism.

I am doing the exercises from one book and learning from the other (not optimal I know), and so I am wondering what definition is considered the convention.

If both are conventionally okay, is something lost by removing the surjective part of the bijection and if so what? I realise it's a rather uninteresting question, but if someone has the time I would appreciate the answer. Thank you!

Answer 1

Consider the two groups $\mathbb{Z}/2$ and $\mathbb{Z}/4$ (the integers modulo $2$ and modulo $4$). We can write $\mathbb{Z}/2=\{0,1\}$ and $\mathbb{Z}/4=\{0,1,2,3\}$.

In modern notation, we would not say that these two groups are isomorphic because there is no bijection between them (as they are different sizes). There is, however, an injective map from $\mathbb{Z}/2$ to $\mathbb{Z}/4$ given by $0\mapsto 0$ and $1\mapsto 2$. In this case, the subgroup $\langle 2\rangle\subseteq\mathbb{Z}/4$ is isomorphic to $\mathbb{Z}/2$.

In older notation, an injective map might be called an isomorphism because the image of the map, in the example above, the image is $\langle 2\rangle$, is isomorphic to $\mathbb{Z}/2$.

Answer 2

The "correct" definition of an isomorphism is a homomorphism which has an inverse homomorphism. For many algebraic structures this is the same as a bijective homomorphism, but not always, see lhf's remark. There is the “Injectivity Implies Surjectivity Trick”. This means that it is often enough for an isomorphism to have an injective homomorphism.