# Math Help - Abstract Algebra: Lagrange's Theorem

1. ## Abstract Algebra: Lagrange's Theorem

Hi all,

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

Problem:

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

Solution:

I said that $|G| = kpq$ for some $k \in \mathbb{Z}_{>0}$. 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

2. Consider $\mathbb{Z}_{pq}, \mathbb{Z}_{2pq}, \mathbb{Z}_{3pq}, ...$ all of these groups have elements of orders $p,q$. While their orders are $pq,2pq,3pq,...$. Thus, it does not seem as there is a way to improve $|G| = kpq$, what your wrote.