# Prove that $\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$ is not isomorphic to any subgroup of $\mathbb{Z}/p\mathbb{Z}$ (with multiplication)

I'm studying hard for my final examination of abstract algebra, but I'm stucked on this question from last year exam.

I'm asked to prove that $G=\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$ is non isomorphic to any subgroup of the multiplicative group $\mathbb{Z}/p\mathbb{Z}$ where $p$ is prime.

If anyone could help me, even just a hint, it would be very appreciated.

Thanks

• Are you meaning the group of invertible elements of $\mathbb{Z}/p\mathbb{Z}$? – egreg Dec 1 '13 at 23:37

The multiplicative group of $\mathbb{Z/pZ}$ is cyclic, any subgroup of a cyclic group is cyclic, and $\mathbb{Z/3Z} \times \mathbb{Z/3Z}$ is not cyclic.