Question

# Let (G,·) be a finite group, and let S be a set with the same cardinality...

Let (G,·) be a finite group, and let S be a set with the same cardinality as G. Then there is a bijection μ:S→G . We can give a group structure to S by defining a binary operation *on S, as follows. For x,y∈ S, define x*y=z where z∈S such that μ(z) = g_{1}·g_{2}, where μ(x)=g_{1} and μ(y)=g_{2}.

First prove that (S,*) is a group.

Then, what can you say about the bijection μ?

is a group as means such that

So that is some element which we define to be

In which case we have

So that the group condition is met and we can say that is also a group

The bijection can therefore be called a group homomorphism as it preserves the group structure and it is a bijection

Therefore, any bijection between a finite set and a group is a group homomorphism

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