Question

Let G be a finite group. Let f and g be prime numbers. Assume that there exists elements in G a and b of orders p and q respectively. Prove that order of G is a multiple of pq

Answer 1

Question 1.

Let $G$ be a finite group. Let $p$ and $q$ be prime numbers. Assume that there exist elements in $G$, say $a$ and $b$, of orders $p$ and $q$, respectively. Prove that order of $G$ is a multiple of $pq$.

Solution.

It follows from Lagrange theorem, that the order of an element of a finite group divides the order of this group. Let the order of $G$ be $n\in\mathbb{N}$. Since $G$ contains $a$ of order $p$ and $b$ of order $q$, then $p$ and $q$ divide $n$. Therefore, $n$ is a multiple of the least common factor of $p$ and $q$. But $p$ and $q$ are relatively prime, so their least common factor is their product $pq$. Thus, $n$ is a multiple of $pq$. ∎