What defines an isomorphism of two groups?

Question

While showing isomorphism of two groups what are the necessary conditions do I need to show, onto property with 1-1 and homomorphism or one-one + showing homomorphism will suffice.

I am getting confused between these two where to show onto and where to not for, e.g., consider two groups $G=(Z,+)$ and $G'=(...., -2m,-m,0,m,2m....., +)$ I define a mapping $f:G \to G' : f(a)=ma$ $\forall a\in G$. Now here in order to show isomorphism do I need to show onto property also or one-one and homomorphism is sufficient

Answer 1

Another way to show that two groups, $G$ and $H$, are isomorphic, is to show that there is a homomorphism $f:G\to H$ such that there exists a homomorphism $g:H\to G$ with

$$f\circ g={\rm id}_H\quad\text{and}\quad g\circ f={\rm id}_G,$$

where ${\rm id}_X$ is the identity map on $X$.

This definition is borrowed from category theory and it illustrates an overarching idea.

In the case of group theory, it is indeed equivalent to showing all of the following:

• $f$ is onto,
• $f$ is one-to-one, and
• $f$ is a homomorphism.

This is a pleasant exercise.