# Abstract Algebra: Lagrange's Theorem

• Mar 16th 2008, 08:47 AM
Jhevon
Abstract Algebra: Lagrange's Theorem
Hi all,

Another problem I am not sure whether I am missing something or not.

Problem:

A finite group $\displaystyle G$ has elements of orders $\displaystyle p$ and $\displaystyle q$, where $\displaystyle p$ and $\displaystyle q$ are distinct primes. What can you conclude about $\displaystyle |G|$?

Solution:

I said that $\displaystyle |G| = kpq$ for some $\displaystyle k \in \mathbb{Z}_{>0}$. As a consequence of Lagrange's Theorem (the orders of the elements must divide |G|)

does that seem like the observation they wanted me to make?

Thanks
• Mar 16th 2008, 09:02 AM
ThePerfectHacker
Consider $\displaystyle \mathbb{Z}_{pq}, \mathbb{Z}_{2pq}, \mathbb{Z}_{3pq}, ...$ all of these groups have elements of orders $\displaystyle p,q$. While their orders are $\displaystyle pq,2pq,3pq,...$. Thus, it does not seem as there is a way to improve $\displaystyle |G| = kpq$, what your wrote.