# Thread: Using Sylow to solve the following:

1. ## Using Sylow to solve the following:

Dear All,

I am stuck on a partucular question which asks the following:

Let G be a group of order pq where p and q are distinct primes. Prove that G is soluble:

I beileve that one method is to use Sylow but I am having trouble solving it. Can someone please assist.

Thanks.

2. Hello.

Recall that a group $\displaystyle G$ is solvable if we can find a series of groups $\displaystyle 0=G_0\leq G_1\leq \ldots \leq G_n=G$ such that $\displaystyle G_i$ is normal in $\displaystyle G_{i+1}$ and $\displaystyle G_{i+1}/G_i$ is abelian.

Suppose that a group $\displaystyle G$ is a group of order $\displaystyle pq$, where $\displaystyle p$ and $\displaystyle q$ are distinct primes (and without loss of generality, suppose $\displaystyle p<q$. By the first Sylow theorem, there are subgroups of $\displaystyle G$ of order $\displaystyle q$. By the third Sylow theorem, the number of of these subgroups $\displaystyle n_q$ divides $\displaystyle p$ and satisfies $\displaystyle n_q\equiv 1\pmod{q}$. We can conclude that $\displaystyle n_q=1$. Take $\displaystyle P$ to be that subgroup. By the second Sylow theorem, $\displaystyle P$ is normal in $\displaystyle G$. We can write the series $\displaystyle 0\leq P\leq G$, noting that $\displaystyle P/0$ is abelian since it is prime order $\displaystyle p$ and $\displaystyle G/P$ is abelian since it is prime order $\displaystyle q$.

This proves it.

3. Alternatively, consider

$\displaystyle 1 \unlhd Z(G) \unlhd G.$

By Lagrange's theorem, $\displaystyle |Z(G)| \in \{1,p,q,pq\}$. If $\displaystyle Z(G)=1, \, pq$, then we're done, so let $\displaystyle |Z(G)|=p$.
Then $\displaystyle |G/Z(G)|=q$. Since $\displaystyle q$ is prime, $\displaystyle G/Z(G)$ is cyclic. The case for $\displaystyle |Z(G)|=q$ is similar.