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**arindamnaskr** Q: Let G, H be cyclic groups, generated by elements x, y. Determine the condition on the orders m,n of x and y so that the map sending x^i --> y^i is a group homomorphism.

Ans: For m<n a function which sends x^i --> y^i can not be well defined; but for m>or=n there is no such problem.

So, the required condition is m > or = n.

AM I RIGHT?

PLEASE HELP

Hi arindamnaskr!

Your condition is true but incomplete.

Suppose we call the map f.

Then for any i, we have

$\displaystyle x^i = x^{m+i}$

Since a map is required to have unique images, that means:

$\displaystyle f(x^i) = f(x^{m+i})$

What can you deduce from that (using the homomorphism property)?