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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

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Solution Summary

This is a proof regarding finite groups.

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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 ...

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