Since [tex]G[/tex] is a group,
• [tex]G[/tex] has an identity element [tex]e[/tex] such that [tex]ae=ea=a[/tex] for all [tex]a\in G[/tex]
• there is an inverse [tex]a^{-1}[/tex] for every [tex]a\in G[/tex] such that [tex]aa^{-1}=a^{-1}a=e[/tex]
• for any [tex]a,b,c\in G[/tex], we have the associative property [tex]a(bc)=(ab)c=abc[/tex]
a) Given that [tex]a^3 b = b a^3[/tex], we have
[tex]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[/tex]
Since [tex]|a|=5[/tex], which means [tex]a^5=e[/tex], we get that
[tex]b = a^2 b a^3[/tex]
Then
[tex]a b = (a^3 b) a^3 = (b a^3) a^3 = b a^6 = b a^5 a = b a[/tex]
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.
