Given that A & B are two subgroups of a group G and let x be in G.

Firstly, Prove that if G is a finite group, then. Moreover, prove that, where the summation is over the elements x chosen one from each disjoint double cosets AxB.

Secondly,

Show that every right coset of A in AxB is of the form Axb, for some b in B. If G is a finite group, then the number of distinct right cosets of A in AxB is