Talk:2.3.12/@comment-160.39.88.238-20120608190643

Writing p for phi: the forward direction of your part b is a bit misleading, in that p inverse is, in fact, p. Bearing this in mind, one can write as parsimonious a proof as you did for your reverse direction.

Assuming the inverse map p(x) = x' is an automorphism of G:

xy = p((xy)') = p(y' x') = p(y')p(x') = yx.