# Cosets

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Note:

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

Â© BrainMass Inc. brainmass.com March 4, 2021, 5:52 pm ad1c9bdddfhttps://brainmass.com/math/linear-algebra/cosets-groups-15178

#### Solution Preview

Proof:

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.

So ...

#### Solution Summary

This is a proof regarding finite groups.

$2.49