anyone please help me this prove..?

Question

Let G be a group and let a,b ϵ G. Show that (a*b) ′ = a′ * b ′ if and only if a* b = b*a.

note: ϵ is an element, ′ is prime.

Answer 1

I know the character " ' " is prime but you don't say what represents in your problem. This might show that you didn't try to solve the problem at all.

"ε is an element" No, ε itself is not an element.
ε is a notation that means "belongs to".
a,b ε G means a,b belong to G or a,b are in G or a,b are elements of G.

Answer 2

Usually the prime
symbol represents the commutator of a and b:
So I will assume that here.
(a*b)'= a*b*a^-1*b^-1.
But then a' makes no sense. You must have 2
elements to define a commutator.
So, what does the prime symbol stand for here?