# Is there any group of order 12 which has a unique Sylow 3-subgroup and unique Sylow 2-subgroup?

Is there any group of order 12 which has a unique Sylow 3-subgroup and unique Sylow 2-subgroup?

In my text, Dummit and Foote, in applications of Sylow theorems, it's mentioned that either if $G$ is a group and $|G|=12$ then $G$ has a normal Sylow 3-subgroup or $G \cong A_4$ (in the later case $G$ has normal Sylow 2-subgroup) and I wonder is "either, or " exclusive or ? or is it inclusive or ?

• A cyclic group of order $12$. – Daniel Fischer Aug 24 '13 at 21:51

Yes, the cyclic group $\mathbb{Z}_{12}$ has unique Sylow $2$- and $3$-subgroups. In general, an abelian group will have a normal (hence unique) Sylow $p$-subgroup for every prime $p$ dividing the group order. Thus, $\mathbb{Z}_6 \times \mathbb{Z}_2$ also has this property, being the (unique) non-cyclic abelian group of order $12$.