I was recently asked this in my abstract algebra class on group actions which seems difficult for me and so need the help on:
Let $ G $ be a group and $ H,K \leq G $ be two subgroups with the groups not necessarily finite. We are to prove for each $ g \in G $ we have the following cardinality of the double coset: $ |HgK| = | K \times (H/H_g) | = | H \times (K/K_g) | $ where we denote: $ H_g = H \cap gKg^{-1} $
Now for the case of finite groups all is OK and basic index arithmetic works but for infinite groups or indices how would I prove equal cardinalities? I should mention I proved $ H \times K $ acts on G via $ (h,k) \cdot g = hgk^{-1} $ and the double coset in question is indeed an orbit so the double cosets form a partition of G but cannot prove the formula. I certainly appreciate the help on proving the formula Thanks