Question

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?

Answer 1

is a group as means such that

So that

Which means

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

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

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