Hi i have completed the answer to this question. Just need your verification on whether it's completely correct or not:

Question:

If G is a group and xEG we define the order ord(x) by:

ord(x) = min{ }

If : G --> H is an injective group homomorphism show that, for each xEG, ord( ) = ord(x)

My answer: Please verify

If = { } then ord( ) = ord(x).

For any integer r, we have x^r = e (or 1) if and only if ord(x) divides r.

In general the order of any subgroup of G divides the order of G. If H is a subgroup of G then "ord (G) / ord(H) = [G:H]" where [G:H] is an index of H in G, an integer.

So order for any xEG divides order of the group. So ord( ) = ord(x)

any suggestions or changes please? thnx