Determine the isomorphism classes of $\operatorname{Aut}(\mathbb{Z}_2\times\mathbb{Z}_3\times\mathbb{Z}_5)$

Question

Determine the isomorphism classes of the automorphism group of the group $\mathbb{Z}_2\times\mathbb{Z}_3\times\mathbb{Z}_5$.

We know that $\gcd(2,3,5)=1$. So $\mathbb{Z}_2\times\mathbb{Z}_3\times\mathbb{Z}_5$ is isomorphic to $\mathbb{Z}_{30}$.

After that how can I proceed?

@user26857 sorry. The actual thing is gcd(2,3)=gcd(2,5)=gcd(3,5)=1. — avijit kundu, Jun 08 '15 at 10:50
I suggest you to include the last comment into the question. — user26857, Jun 08 '15 at 15:39

score 0 · Answer 1 · answered Jun 08 '15 at 07:18

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$\textrm{Aut}(\mathbb{Z}_n) \cong \mathbb{Z}_n^{\times}$

answered Jun 08 '15 at 07:18

Mr.Fry

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score 0 · Answer 2 · answered Jun 08 '15 at 07:21

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Well as you observed, since $2,3,$ and $5$ are pair-wise co-prime, $$\text{Aut}(\mathbb{Z}/2 \times \mathbb{Z}/3 \times \mathbb{Z}/5)\simeq \text{Aut}(\mathbb{Z}/2)\times \text{Aut}(\mathbb{Z}/3) \times \text{Aut}(\mathbb{Z}/5).$$ Can you continue from here?

answered Jun 08 '15 at 07:21

M. L. Nguyen

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I can't see where exactly the OP observed that $2$, $3$, and $5$ are pair-wise coprime. – user26857 Jun 08 '15 at 08:43

Determine the isomorphism classes of $\operatorname{Aut}(\mathbb{Z}_2\times\mathbb{Z}_3\times\mathbb{Z}_5)$

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