# Math Help - If a is an element of order m in a group G and a^(k) = e, prove that m divides k.

1. ## If a is an element of order m in a group G and a^(k) = e, prove that m divides k.

If a is an element of order m in a group G and a^(k) = e, prove that m divides k.

2. Do a long division on the exponent.

3. Originally Posted by FancyMouse
Do a long division on the exponent.

I don't understand what you mean. Can you explain?

4. Originally Posted by rainyice
I don't understand what you mean. Can you explain?
Let k=rm+q where r is an integer and 0<=q<m. What can you say about $a^q$?

5. Originally Posted by FancyMouse
Let k=rm+q where r is an integer and 0<=q<m. What can you say about $a^q$?
it has infinite order?

6. $k=rm+q$, where $q.

So, $1=a^k=a^{rm+q}=(a^{m})^r\cdot a^q = a^q$

Since $q, what is the only such $q$ that gives $a^q = 1$?