1. Orders of Groups

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{ $r \geq 1: x^r = 1$}

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

If $\theta(x)$ = { $x^r: r \epsilon Z$} then ord( $\theta(x)$) = 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( $\theta(x)$) = ord(x)

any suggestions or changes please? thnx

2. If ord(x)=a then $\left[ {\phi (x)} \right]^a = \left[ {\phi (x^a )} \right] = \phi (e) = e'$.
Now suppose that $ord\left[ {\phi (x)} \right] = b < a$.
Then
$\left[ {\phi (x)} \right]^b = e' = \left[ {\phi (x)} \right]^a$
$\phi (x^b ) = \phi (x^a )$
$x^b = x^a$ (injective)
$x^{a - b} = e$

i) $(\theta(x))^a = e'$
ii) $0 < b < a \implies (\theta(x))^b \neq e'$.
therefore the contradiction seems to arise from the fact that if $x^{a} = x^{b}$, then $\left[ {\phi (x)} \right]^b = e'$ which is not what statement (ii) says.