Thread: 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.

Hello.

Recall that a group is solvable if we can find a series of groups such that is normal in and is abelian.

Suppose that a group is a group of order , where and are distinct primes (and without loss of generality, suppose . By the first Sylow theorem, there are subgroups of of order . By the third Sylow theorem, the number of of these subgroups divides and satisfies . We can conclude that . Take to be that subgroup. By the second Sylow theorem, is normal in . We can write the series , noting that is abelian since it is prime order and is abelian since it is prime order .

This proves it.

Alternatively, consider



By Lagrange's theorem, . If , then we're done, so let .
Then . Since is prime, is cyclic. The case for is similar.