This is a very strangely formulated problem. E.g., it goes from x, y to some unrelated a, b...Show that the following are equivalent. (A,~) is a normal group. Let x,y in set A.

a) xyx'y = i

b) (xy)^2 = a^2 b^2

c) A is commutative

First: what is a "normal group"? Is "normal" used in the usual sense or is it a technical term?

The only way I can interpret the problem is the following. Please correct me if I am wrong.

Given a group , prove that the following are equivalent.

(1) For all , (group unit)

(2) For all ,

(3) is commutative.

If this is the case, then this claim is proved by multiplying both sides of an equation on the right or on the left. In general, if are elements of some group , then the following three facts are equivalent:

(a)

(b)

(c)

For example, if , then multiplying both sides on the right we have . Conversely, if , then multiplying both sides on the right by we get . It is very important to multiply either both sides on the left or both sides on the right.

So, for your problem, suppose that . Multiply this equation by on the right. Note that to deduce that is commutative, you must know thatfor alland .