Prove that the given group is an Abelian Group

Question

Let G be a group in which $(a*b)^3=a^3 * b^3$ and $(a*b)^5=a^5*b^5$ for all $a,b \in G$. Show that G is an Abelian group ?

Answer 1

Since $$(ab)^3 = a^3b^3 \Longrightarrow (ba)^2 = a^2b^2;\;\;\;\;(*)$$

and $$(ab)^5 = a^5b^5 \Longrightarrow (ba)^4 = a^4b^4;\;\;\;\;$$

so $$ (a^2b^2)^2 = ((ba)^2)^2= a^4b^4\Longrightarrow b^2a^2 = a^2b^2$$

using $(*)$ we get $$b^2a^2=(ba)^2 \Longrightarrow ba = ab$$

Answer 2

First observe that $(ab)(ab)(ab)=a^3b^3\Rightarrow a^2b^2=(ba)(ba)=(ba)^2$

So,

$(ab)(ab)(ab)(ab)(ab)=a^5b^5$

$\Rightarrow$ $a^3b^3(ab)(ab)=a^5b^5\Rightarrow$ $b^3(ab)(ab)=a^2b^5$

$\Rightarrow$ $b^3(ab)(ab)=(a^2b^2)b^3$

$\Rightarrow$ $b^3(ab)(ab)=(ba)(ba)b^3$

$\Rightarrow$ $b^2(ab)(ab)=abab^3$

$\Rightarrow$ $b^2(aba)=abab^2$

$\Rightarrow$ $b(ba)(ba)=abab^2$

$\Rightarrow$ $ba^2b^2=abab^2$

$\Rightarrow$ $ba^2=aba$

$\Rightarrow$ $ba=ab$.