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.