Show that if every element of a group G is its own inverse t

Show that if every element of a group (G,.) is its own inverse, then G is an abelian group.

Solution

Suppose x = x^-1 for all x G

=> x² = x(x) = x(x^-1) = e , for all x G.

Now if x,y G, then so is xy G ,

=> (xy)² = e,

=> (xy)(xy) = xyxy = e.

=> Since , xy = (xe)y = x(xyxy)y = (xx)yx(yy) = eyxe = yx,

=> xy = yx

=> G is abelian.