AnswerBun.com

Showing that a group $G$ such that 3 does not divide $|G|$ is Abelian.

Mathematics Asked by user778657 on December 21, 2020

I asked this question here Understanding the hint of a question to show that $G$ is Abelian. but I did not receive answers to all my questions. So I am attaching a trial to the solution of the question which I found online and could not fully understand it.

First

Here is the main question: Let $G$ be a finite group such that 3 does not divide $|G|$ and such that the identity $(xy)^3 = x^3 y^3$ holds for all $x,y in G.$ Show that $G$ is abelian.

And here is the hint I got for the question:

First show that the map $G rightarrow G$ given by $x mapsto x^3$ is bijective. Then show that $x^2 in Z(G)$ for all $x in G.$

And here is a trial for the solution:

enter image description here

enter image description here

enter image description here

My questions are:

1-should we show that the map $G rightarrow G$ given by $x mapsto x^3$ is bijective because of the following step in the solution above:

$$(y^2x^2)^3 = (x^2y^2)^3 implies (y^2x^2) = (x^2y^2)$$ because there is no element in $G$ with order $3.$?

2- Also, how the given solution showed that $x^2 in Z(G)$ for all $x in G$? And can this statement be restated as $x^2 in Z(G)$ for all $y in G$?

3- Is the given trial of the solution correct? if not, how can we correct it?

Could anyone help me in answering those questions, please?

One Answer

Given condition says $x mapsto x^3$ is a group homomorphism, it is injective because its kernel consists of elements of order $3$. It is then bijective by comparing size.

$xy^3x^{-1} = (xyx^{-1})^3 = x^3y^3x^{-3}$, hence $x^2$ commutes with $y^3$, and $y^3$ can be any element in $G$, so $x^2$ is in center for any $x in G$.

Then $x^3y^3 = (xy)^3 = x(xy)^2y$, i.e. $xy=yx$, $G$ is Abelian.

the given trial seems correct btw.

Correct answer by yisishoujo on December 21, 2020

Add your own answers!

Related Questions

Linear independence in $mathbb{Z}_2$ and $mathbb{R}$

1  Asked on February 4, 2021 by andrew-yuan

 

How to denote a FOR…LOOP in mathematics

1  Asked on February 3, 2021 by user3190686

 

Drawing numbers from an interval

1  Asked on February 3, 2021 by itsme

 

Projective bundle and blow up

0  Asked on February 3, 2021 by konoa

   

Calculating the integral.

0  Asked on February 2, 2021 by 1_student

       

Ambiguity in delta equal sign

1  Asked on February 2, 2021

   

Ask a Question

Get help from others!

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