Note: C means set containment (not proper set containment), |G : K| means index of subgroup K in G, and G # K means K is a normal subgroup of G
Let K C H C G be groups, where K # G and |G : K| is finite. Show that |G/K : H/K| is also finite and that |G/K : H/K|=|G : H|
Since KCHCG and |G:K| is finite, then |G:H| is finite. K is a normal subgroup of G, then H/K is a subgroup of G/K. Suppose |G:H|=n, then we have
G=g1H U g2H U...U gnH, where U ...
A proof involving subgroups and indicies is offered. The proof is concise.