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March 21st, 2009, 11:00 AM   #1
Joined: Mar 2009

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Double Cosets

Let and be subgroups of the group . For each define the double coset of in to be the set
a) Prove that is the union of the left cosets where is the orbit containing of acting by left multiplication on the set of left cosets of .
b) Prove that is a union of right cosets of .
c) Show that and are either the same set or are disjoint for all . Show that the set of double cosets partitions .
d) Prove that .
e) Prove that .

This is, well, exercise 4.1.10 from Dummit & Foote. It doesn't seem too difficult, but for some reason, I can't seem to do it.
Independent studies are quite frustrating at times. Sigh.
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cosets, double

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