My Math Forum Double Cosets

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 March 21st, 2009, 11:00 AM #1 Newbie   Joined: Mar 2009 Posts: 21 Thanks: 0 Double Cosets Let $H$ and $K$ be subgroups of the group $G$. For each $x \in G$ define the $HK$ double coset of $x$ in $G$ to be the set $HxK=\{hxk | h \in H, k \in K\}$. a) Prove that $HxK$ is the union of the left cosets $x_1K,...,x_nK$ where $\{x_1K,...,x_nK\}$ is the orbit containing $xK$ of $H$ acting by left multiplication on the set of left cosets of $K$. b) Prove that $HxK$ is a union of right cosets of $H$. c) Show that $HxK$ and $HyK$ are either the same set or are disjoint for all $x,y \in G$. Show that the set of $HK$ double cosets partitions $G$. d) Prove that $|HxK|=|K||H : H \cap xKx^{-1}$. e) Prove that $|HxK|=|H||K : K \cap x^{-1}Hx|$. 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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