# Thread: Show x^n is discrete in {x^k}

1. ## Show x^n is discrete in {x^k}

Hmmm...No listing for Group Theory. I'll post it here for now.

If x is an element of infinite order in a group G, prove that the elements $\displaystyle x^n \text{, }n \in \mathbb{Z}$are all distinct.
I don't even have an idea where to start. Can anyone give me a push in the right direction?

Thanks!
-Dan

2. ## Re: Show x^n is discrete in {x^k}

Originally Posted by topsquark
Hmmm...No listing for Group Theory. I'll post it here for now.
I don't even have an idea where to start. Can anyone give me a push in the right direction
Suppose that $\displaystyle j\ne k$ but $\displaystyle x^j=x^k$ can $\displaystyle x$ be of infinite order?

3. ## Re: Show x^n is discrete in {x^k}

Originally Posted by Plato
Suppose that $\displaystyle j\ne k$ but $\displaystyle x^j=x^k$ can $\displaystyle x$ be of infinite order?
I have a thought. Is this where you were going with it?
Let k > j. Suppose that we have

$\displaystyle x^j = x^k$

Then
$\displaystyle x^{-j} x^j = x^{-j}x^k$

$\displaystyle 1 = x^{k - j}$

So |x| is of finite order k - j, a contradiction.

-Dan