# Thread: 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 $\displaystyle a^q$?

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

6. $\displaystyle k=rm+q$, where $\displaystyle q<m$.

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

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