Let G be a group and H ≤ G be a subgroup. Let G/H be the set of left cosets {xH | x ∈ G} and let G act on G/H through left multiplication, i.e., g · (xH) = (gx)H. Show that if N is a normal subgroup of G and N ≤ H, then every element n ∈ N acts trivially on G/H (i.e., fixes every element in G/H).
