Let G be a group acting on the sets X$,$ Y $.$ A map f : X $->$ Y is called Gequivariant if f$($g $$ x$)=$ g $$ f$($x$)$ for all g in G$,$ x in X$.$

$($a$)$ Show that the composition of G$-$equivariant maps is again G$-$equivariant.

$($b$)$ Show that if a G$-$equivariant map f is a bijection, then the inverse f

$1$

is

also G$-$equivariant. In this case we call the actions on X and Y equivalent.