Understanding the hint of a question to show that $G$ is Abelian.

Here is the question I want to answer:

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 $$x in G.$$

My questions are:

1- Could anyone explain for me why we should show that the map $$G rightarrow G$$ given by $$x mapsto x^3$$ is bijective?

2- Also, why we should show that $$x^2 in Z(G)$$ for all $$x in G$$?

3- And how proving the hint will show that $$G$$ is Abelian?

Mathematics Asked on January 2, 2021

Related Questions

How to prove that there is not a monomorphism from Klein 4-group to $Z_6$(or a epimorphism from $Z_6$ to $V_4$)?

1  Asked on January 25, 2021 by tota

Homomorphisms between fields are injective.

4  Asked on January 25, 2021

We have an integer n. We have n boxes where each box contains a non-negative amount of balls. Find all the permutations which satisfy some criteria

1  Asked on January 25, 2021 by michael-blane

Approximation of the sum of a series $S(t)=-frac{2}{pi t} cos(frac{pi t}{2}) sum_{m odd}^{infty}frac{m^2alpha_m}{t^2-m^2}$ as $tto +infty$

0  Asked on January 25, 2021 by yolbarsop

Homogenous space of elliptic curve E/$Bbb Q$

1  Asked on January 25, 2021 by bellow

On the functional square root of $x^2+1$

9  Asked on January 25, 2021 by user1551

Is there a simple way to find all the solutions of $x_1 + x_2 + dots + x_k + dots + x_K = N$ when $x_k$s and $N$ are all non-negative integers?

2  Asked on January 25, 2021 by cardinal

Minimizing the Schatten 1-norm over symmetric matrices.

2  Asked on January 25, 2021 by gene

The frontier of a set

1  Asked on January 25, 2021 by gal-ben-ayun

express a matrix using Kronecker product

0  Asked on January 25, 2021 by jyothi-jain

Approximating multiples of reals with integers

1  Asked on January 25, 2021

How is this sentence written correctly in maths? (sets)

0  Asked on January 25, 2021 by eyesima

Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?

6  Asked on January 24, 2021 by user713999

Partitioning $mathbb{R}$ for given functions.

0  Asked on January 24, 2021

How to count the number of symmetries of a 3-d object?

1  Asked on January 24, 2021

Is checking really needed?

3  Asked on January 24, 2021 by 1b3b

Struggling to understand proof, there is no simple group of order $525$

1  Asked on January 24, 2021 by scott-frazier

prove the following compact result about $GL(n,mathbb{R})$

1  Asked on January 24, 2021

i need to proove that An∪B→A∪B

1  Asked on January 24, 2021 by methodcl

How can I show that $T(omega) = T(overline{omega})$ when $X_{t}(omega)=X_{t}(overline{omega})$ for all $t in [0,T(omega)]cap [0,infty)$

1  Asked on January 24, 2021 by minathuma