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March 21st, 2009, 10:00 AM   #1
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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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