Question

Please help me with the below question.

Answer 1

Since $G$ is a group,

• $G$ has an identity element $e$ such that $ae=ea=a$ for all $a\in G$

• there is an inverse $a^{-1}$ for every $a\in G$ such that $aa^{-1}=a^{-1}a=e$

• for any $a,b,c\in G$, we have the associative property $a(bc)=(ab)c=abc$

a) Given that $a^3 b = b a^3$, we have

$a^3 b = b a^3 \implies a^2 (a^3 b) = a^2 (b a^3) \implies a^5 b = a^2 b a^3$

Since $|a|=5$, which means $a^5=e$, we get that

$b = a^2 b a^3$

Then

$a b = (a^3 b) a^3 = (b a^3) a^3 = b a^6 = b a^5 a = b a$

as required.

b) What do you mean by "rank of an element in a group"? That doesn't sound like a standard concept as far as I know.