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I'm asked to describe $Aut(C_{21}),Aut(C_{24})...$ as a product of cyclic groups - but I'm wondering is there a general way to do this?

automobile1
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1 Answers1

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An automorphism of $\mathbb{Z}/n\mathbb{Z}$ is uniquely determined by the image of $1$ (generator), and this one as to be a generator, hence an invertible element.

Hence, you get a bijection $\mathrm{Aut}(\mathbb{Z}/n\mathbb{Z})\to (\mathbb{Z}/n\mathbb{Z})^*$, which is in fact a group isomorphism. In some cases (for example when $n$ is prime) then the group is cyclic, but not always, like for example when $n=8$.

Jérémy Blanc
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  • sorry what is the notation $\mathbb{Z} / n \mathbb{Z}$? – automobile1 Jan 26 '15 at 23:13
  • @automobile1 see http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n there you also find an explict description of the structure in all cases. – quid Jan 26 '15 at 23:18