which one of these may not be abelian? If a group G has these orders. which one of these may not be abelian?
4,31,55,39 and 121
since 4 and 121 is prime square. they are abelian. and 31 is prime therefore cyclic so abelian. what about 55 and 39? can we say $55=5.11$ and $11\equiv1$ (mod 5)  $39=3.13$ and $13\equiv1$ (mod 3)  they may not be cyclic. 
but my assumption seems wrong. if it is not cyclic we cant say it is not abelian.
 A: Using Sylow theorems you can observe that a group of order $pq$ where $p$, $q$ are primes, $p < q$, has a subgroup of order $p$ and a unique subgroup of order $q$. Hence, the subgroup of order $q$ is normal, and our group is a semidirect product of these subgroups. So if there is a nontrivial homomorphism $\mathbb{Z}_p \to Aut(\mathbb{Z}_q)$, the corresponding semidirect product is not abelian.
A: The structure of groups of order $pq\enspace(p<q\text{ primes})$ is known:


*

*If $q\not\equiv 1\mod p$: $G$ is cyclic, isomorphic to $\mathbf A/pq\mathbf Z$.

*If $q\equiv 1\mod p$: either $G$ is cyclic as above, or $G$ is not abelian, and it is isomorphic to a semi-direct product $\;\mathbf Z/q\mathbf Z\rtimes_\theta\mathbf Z/p\mathbf Z$, where $\:\theta\colon\mathbf Z/p\mathbf Z\to(\mathbf Z/q\mathbf Z)^\times$. 


One shows all non-trivial morphisms $\theta$ give isomorphic semi-direct products. Hence there are only $2$ non-isomorphic group structures of order $pq$.
In addition, we may note, in the case $p=2$, that the non-abelian group structure is isomorphic to the dihedral group $D_q$ (or $D_{2q}$, depending on the notation).
