Hi,
i was wondering if anyone could guide me through this question:
a) Suppose G is an abelian group, and let f:G->G be defined by f(x)=x^3
i) show that f is an isomorphism if |G| is not divisible by 3.
b) give an example to show that f:x->x^3 need not be a homomorphism if G is non-abelian.
c) suppose that G is a group in which x^2=e for all x in G. Prove that G is abelian.
Thanks!
[quote=1234567;482325]The question implies the group is finite (the result does not holds otherwise). If is finite and is an injection, then it holds that is a surjection too.
Can you see why this is? Essentially, it is because of the orders. You know that the pre-image and the image have the same order, and every element of the pre-image maps to a unique element of the image. So it really has to be a surjection. (The converse also holds).