Prove that xy1 x1 y1 for all x y elementof iff is abelian g

Prove that (xy)^-1 = x^-1 y^-1 for all x, y elementof iff is abelian group.

Solution

From the group axioms it follows that (xy)(y ^1x^ 1 ) = x(y(y ^1x^ 1 )) = x((yy^1 )x^ 1 ) = x(ex^1 ) = xx^1 = e. Similarly (y ^1x^ 1 )(xy) = e, and thus y ^1x^ 1 is the inverse of xy, as required.

Note in particular that (x ^1 )^ 1 = x for all elements x of a group G, since x has the properties that characterize the inverse of the inverse x^ 1 of x. Given an element x of a group G, we define x^n for each positive integer n by the requirement that x ^1 = x and x^ n = (x ^n1) *x for all n > 1. (This is an example of a so-called inductive definition, where some quantity u(n) is defined for all positive integers n by specifying u(1) and also the rule that determines u(n) in terms of u(n 1) for each n > 1.) We also define x ^0 = e, where e is the identity element of the group, and we define x^ n to be the inverse of x^ n for all positive integers n.