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

• Apr 2nd 2010, 09:58 PM
rainyice
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.
• Apr 3rd 2010, 12:30 AM
FancyMouse
Do a long division on the exponent.
• Apr 3rd 2010, 01:42 PM
rainyice
Quote:

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

I don't understand what you mean. Can you explain?
• Apr 3rd 2010, 05:00 PM
FancyMouse
Quote:

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 $\displaystyle a^q$?
• Apr 3rd 2010, 05:12 PM
rainyice
Quote:

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?
• Apr 3rd 2010, 06:10 PM
chiph588@
$\displaystyle k=rm+q$, 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 $\displaystyle a^q = 1$?