# Problem related to semidirect product

I have a small question regarding the semidirect product. Consider a group $$G$$ which is the semidirect product $$\mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5)$$ (internal semidirect product). Let $$\phi\colon\mathbb{Z}_3\to\operatorname{Aut}(\mathbb{Z}_5 \times \mathbb{Z}_5)$$. There will be $$\phi_0 ,\phi_1 ,\phi_2$$ corresponding to $$\bar{0},\bar{1},\bar{2}$$ of $$\mathbb{Z}_3$$, right?

• Can you compute the automorphism group of $\Bbb Z_5 \times \Bbb Z_5$? A fact you will need once you have done that is that the image of a generator of $\Bbb Z_3$ must have order dividing the order of the codomain. – Lukas Oct 11 '18 at 9:45
• I don't understand. Do you mean that this semidirect product can not exist? – Buddhini Angelika Oct 11 '18 at 9:49

## 1 Answer

Since Aut$$(\mathbb{Z}_5)=C_4$$, Aut($$\mathbb{Z}_5 \times \mathbb{Z}_5$$) $$\cong C_4\times C_4$$. The only action $$\mathbb{Z}_3$$ can have on $$\mathbb{Z}_5 \times \mathbb{Z}_5$$ is the trivial one. Thus you get a direct product in the end.

• Thanks a lot @Jianping but when I compute automorphisms of order 3 in the automorphism group of $\mathbb{Z}_5 \times \mathbb{Z}_5$, in GAP i get several automorphisms... – Buddhini Angelika Apr 17 '19 at 16:11