Number of non isomorphic groups of order 10

Find the number of non-isomorphic groups of order 10.

I tried using fact that since there are no non abelian groups of order 10, so we are left with abelian groups which will be $\Bbb Z_2 \times\Bbb Z_5$ and $\Bbb Z_{10}$. Is this reasoning ok?

What if we had some tougher problem. Here order was simple, just 10. I know there is no general formula to find total number of non-isomorphic groups of order $n$. But from an exam point of view, how to deal with such questions?

• $\mathbb Z_2 \times \mathbb Z_5 \cong \mathbb Z_{10}$ – Namaste Jan 15 '15 at 17:17
• Z_2xZ_5 and Z_10 are isomorphic. Anyways do u know sylow's theorem? – Jack Yoon Jan 15 '15 at 17:17
• And there is a non Abelian group of order 2p – Jack Yoon Jan 15 '15 at 17:18
• No I dont know sylows theorm yet. And don't we have S_3 which has order 6 and next non ableian group comes at order 24. So 10 order doesn't have any nonabelian group.? – ketan Jan 15 '15 at 17:23
• There exists a non-abelian dihedral group $D_n$ of order $2n$ for any $n>2$. It is not difficult to show that $D_5$ is the only non-abelian group of order 10 using Cauchy's theorem (i.e. the result that in a group of order 10 there necessarily is an element of order 5). – Jyrki Lahtonen Jan 15 '15 at 17:55