Consider the groups $G = \{0,1,2\} = \mathbb Z_3$ and $H = \{a,b,c\}$
given by the following multiplication tables:
The first one isn't really multiplication but in my notes it said it doesn't really matter.
So how do I show an isomorphism? The groups have the same size so they can be bijective right? But it just seems so abstract to show if there's an isomorphism… Exactly what do we have to check.
Best Answer
First, find the identity in each. In your first example, it’s “$0$”, while in the second it’s “$b$”. How to see this? It’s the element whose row and column matches the labeling rows. Now see whether you can match some nonidentity element in the first example to a nonidentity element in the second so that you force a homomorphism.