1. ## Group Theory Proof

Can somebody help me with this proof please?

Suppose that G is a group with elements a and b of orders p and q respectively, where p and q are distinct primes.

Suppose also that for some integers i and j, that a^i = b^j. Show that p|i and q|j.

2. Hi
Originally Posted by Cairo
Suppose that G is a group with elements a and b of orders p and q respectively...
This means that $\displaystyle a^p=b^q=e$ where $\displaystyle e$ is the identity element of the group.
... where p and q are distinct primes.

Suppose also that for some integers i and j, that a^i = b^j. Show that p|i and q|j.
You may try using $\displaystyle (a^i)^p=(b^j)^p$ and $\displaystyle (a^i)^q=(b^i)^q$.

3. could be having a dumb moment....but I'm still not following.

4. As $\displaystyle (a^i)^p=a^{ip}=(a^p)^i=e^i=e$ and $\displaystyle a^{ip}=b^{jp}$, we get that $\displaystyle e=b^{jp}$ which requires that there exists $\displaystyle n_1\in\mathbb{N}$ such that $\displaystyle jp = n_1 q$. What can be deduced from this ?

5. that p|i ?

6. Bad luck, it's the other one... (it can be shown acknowledged that if $\displaystyle u|vw$ then $\displaystyle u|v$ or $\displaystyle u|w$)