# subgroups with prime factors [duplicate]

The order of a group $G$ is $pq$ where $p$ and $q$ are both prime. Show that all proper subgroups of $G$ are cyclic. I need help with this. I know that all the orders of subgroups divide the order of the larger group, but I do not know where to go from there.

## marked as duplicate by DonAntonio, Dan Rust, Lord_Farin, Nick Peterson, Dennis GulkoNov 12 '13 at 18:39

The order of a subgroup $H$ of $G$ must divide the order of $G$, then, if $H$ is a proper subgroup of $G$, its order could be only $p , q$ or 1

Observing that a group of prime order is cyclic you can prove the assert.

Consider the following:

What are the divisors of $pq$?

What kind of groups exist for the orders less than $pq$?

• I think you mean subgroups. Certainly there are many groups of order less than $pq$ that have nothing to do with $G$. – mathematics2x2life Nov 12 '13 at 17:34
• There aren't many groups of order less than $\;pq\;$ and whose order is a divisor of $\;pq\;$ , as explained in the second line... – DonAntonio Nov 12 '13 at 17:42

The order of a subgroup $H\leq G$ must divide the order $G$. Therefore $|H|$ is either $1$, $p$, $q$ or $pq$. If $H$ is a proper subgroup of $G$ (that is $H$ is not equal to $G$) then it must be the trivial group, or it must be of prime order. Show that in this case it must be cyclic.