Let r be a subgroup of G.
Let Ar be the set of all products (finite) of elements of the form (gi ri gi^-1) or (gi ri^-1 gi^-1) i = 1 to n.
How can one show that Ar is normal in G.
Isn't it clear this question belongs in "Abstract Algebra and Linear Algebra"??
Anyway, this is trivial when one realizes that . The set that you denoted Ar is more usually denoted by and it's called
the normal closure of the subgroup r in G.
It also way more usual to denote sets, groups, subgroups, etc. with capital letters...