In: Advanced Math
Let f be a group homomorphism from a group G to a group H
If the order of g equals the order of f(g) for every g in G must f be one to one.
be a group homomorphism .
Suppose
be two elements
. Now ,
, where
is the identity elemnt of H .
, As f is a homomorphism .
As given order of
equals to
so order of
is equals to order of
Order of
= Order of
Now as order of the identity element is always 1 so ,
Order of
is 1 .
As only identity can have order 1 so ,
So
implies
.
Hence
is one-to-one.
.
.
If you have any doubt or need more clarification at any step please comment.