C means set containment (not proper)
|G:H| means index of subgroup H in G
U means union of sets
E means belonging to
Let K C H C G be groups. Show that both |G:H| and |H:K| are finite if and only if |G:K| is finite, and then |G:K| = |G:H||H:K|.
Hint: if |H:K| = n, let Kh1, Kh2, ..., Khn be the distinct cosets of K in H. Show that Hg = Kh1g UKh2g U ......U Khng is a disjoint union for all g E G© BrainMass Inc. brainmass.com November 24, 2022, 11:42 am ad1c9bdddf
If |G:K| is finite and |G:K|=|G:H||H:K|, then obviously |G:H| and |H:K| is finite.
If |G:H| and |H:K| is finite, then we can suppose |G:H|=m, |H:K|=n.
This is a proof regarding finite groups.