Can we predict something about the automorphism of an abelian group of order $315$?

Asked Sep 24 '21 at 11:59

Active Sep 24 '21 at 21:01

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Can we predict something about the automorphism of an abelian group of order $315$?

I know that $ \mbox{Aut}(\mathbb{Z_3} \times \mathbb{Z_3}) = \mbox{GL}_2(\mathbb{Z_3})$ and $|\mbox{Aut}(\mathbb{Z_9})| = \varphi(9)$ where $\varphi(9)$ is the euler toitent's function.

Let $G$ be an abelian group of order $315$ then $G \cong \mathbb{Z_3} \times \mathbb{Z_3} \times \mathbb{Z_7} \times \mathbb{Z_5}$ or $G \cong \mathbb{Z_9} \times \mathbb{Z_7} \times \mathbb{Z_5}$

Now in the second case I guess $Aut(G) \cong (\mathbb{Z_9})^* \times \mathbb{Z_7}^* \times \mathbb{Z_5}^*$

What about the first case?

edited Sep 24 '21 at 21:01

Nicky Hekster

49,281

asked Sep 24 '21 at 11:59

ThirstForMaths

1,972

1

In the first case $Aut(G)$ is again the direct product of the automorphism groups. This is a simple exercise since the Sylow $p$-subgroups are unique. – David A. Craven Sep 24 '21 at 12:02
I should add this duplicate for clarity. – Dietrich Burde Sep 24 '21 at 12:10
Note that $GL_2(\Bbb F_3)$ has $48$ elements, but $(\Bbb Z/3)^\times (\Bbb Z/3)^$ has only $4$ elements. So this cannot be equal. – Dietrich Burde Sep 24 '21 at 12:17
@DietrichBurde $Aut(G \times H) \cong Aut(G) \times Aut(H)$ but $\mathbb{Z_3}$ ,$ \mathbb{Z_3}$ are not groups of mutually prime order. – ThirstForMaths Sep 24 '21 at 12:18
I don't understand the point of closing this question, the post associated also doesn't make sense.I have no idea of endomorphism stuffs – ThirstForMaths Sep 24 '21 at 12:19
Do you still have a question now? – Dietrich Burde Sep 24 '21 at 12:21
Yes, I do .So what is Aut(G) in the first case is it isomorphic to $Aut(\mathbb{Z}_3 \times \mathbb{Z}_3) \times Aut(\mathbb{Z}_7) \times Aut(\mathbb{Z}_5)$? – ThirstForMaths Sep 24 '21 at 12:23
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Yes, right, exactly as you said in your post, and what the duplicate is saying. It is isomorphic to $GL_2(\Bbb F_3)\times {\rm Aut}(\Bbb Z/7)\times {\rm Aut}(\Bbb Z/5)$. – Dietrich Burde Sep 24 '21 at 12:24
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Is $Aut(\mathbb{Z}_3 \times \mathbb{Z}_3) \times Aut(\mathbb{Z}_7) \times Aut(\mathbb{Z}_5)$ isomorphic to $Gl_2(\mathbb{Z_3}) \times Aut(\mathbb{Z}_7) \times Aut(\mathbb{Z}_5)$ ? – ThirstForMaths Sep 24 '21 at 12:24
Okay I understand now... – ThirstForMaths Sep 24 '21 at 12:25
Very good, I am glad I could help. – Dietrich Burde Sep 24 '21 at 12:27

Can we predict something about the automorphism of an abelian group of order $315$?

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