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.


  • 1
    $\begingroup$ Are you meaning the group of invertible elements of $\mathbb{Z}/p\mathbb{Z}$? $\endgroup$ – 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.


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