Mathematics Asked by La Rias on December 24, 2020

Let $ G $ and $ H $ be finite groups. If $ gcd(|G|,|H|) = 1 $, then I want to show that

$$

text{Aut}(G times H) cong text{Aut}(G) times text{Aut}(H).

$$

In my attempt, I first defined the map $ f: text{Aut}(G) times text{Aut}(H) to text{Aut}(G times H) $ by

$$

forall (phi_{G},phi_{H}) in text{Aut}(G) times text{Aut}(H): quad

f(phi_{G},phi_{H}) stackrel{text{def}}{=} phi_{G} times phi_{H},

$$

but I don’t know how to use the fact that $ gcd(|G|,|H|) = 1 $.

Automatically ${rm Aut}(G)times{rm Aut}(H)subseteq{rm Aut}(Gtimes H)$. The gcd condition forces this to be an equality.

Let $phiin{rm Aut}(Gtimes H)$. Show $phi(Gtimes1)= Gtimes1$; argue $subseteq$ by contradiction (show the order of an element outside of $Gtimes1$ is divisible by a factor of $|H|$...). Symmetrically, $phi(1times H)=1times H$.

After that it is straightforward to check $phi=(phi|_G,phi|_H)$, hence $phiin{rm Aut}(G)times{rm Aut}(H)$.

Correct answer by anon on December 24, 2020

0 Asked on October 5, 2020 by aulwmate

derivatives finite difference methods finite differences numerical methods

1 Asked on October 5, 2020 by user378298

3 Asked on October 5, 2020 by xxx

0 Asked on October 3, 2020 by blm

1 Asked on October 3, 2020 by benjamin

abstract algebra automorphism group finite groups group theory

1 Asked on October 2, 2020 by stf

2 Asked on October 2, 2020 by kantura

1 Asked on October 1, 2020 by vignesh-sk

integration multiple integral multivariable calculus physics spheres

1 Asked on October 1, 2020 by calmat

1 Asked on September 30, 2020 by jos-victor-gomes

1 Asked on September 30, 2020 by laufen

1 Asked on September 29, 2020 by ranveer-masuta

1 Asked on September 29, 2020 by mizerex

0 Asked on September 29, 2020 by roulette01

1 Asked on September 27, 2020 by androidv11

0 Asked on September 25, 2020 by integrand

2 Asked on September 23, 2020 by objectivesea

Get help from others!

Recent Answers

- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP