DoubleCoset 2006-10-01 11:26:31 2006-10-06 04:39:47 Definition \def\genby#1{\langle#1\rangle} \PMlinkescapeword{coset} \PMlinkescapeword{cosets} \PMlinkescapeword{obvious} Let $H$ and $K$ be subgroups of a group $G$. An \emph{$(H,K)$-double coset} is a set of the form $HxK$ for some $x\in G$. Here $HxK$ is defined in the obvious way as \[ HxK = \{ hxk \mid h\in H \hbox{ and } k\in K \}. \] Note that the $(H,\{1\})$-double cosets are just the right cosets of $H$, and the $(\{1\},K)$-double cosets are just the left cosets of $K$. In general, every $(H,K)$-double coset is a union of right cosets of $H$, and also a union of left cosets of $K$. The set of all $(H,K)$-double cosets is denoted $H\backslash G/K$. It is straightforward to show that $H\backslash G/K$ is a \PMlinkname{partition}{Partition} of $G$, that is, every element of $G$ lies in exactly one $(H,K)$-double coset. In contrast to the situation with ordinary \PMlinkname{cosets}{Coset}, the $(H,K)$-double cosets need not all be of the same cardinality. For example, if $G$ is the \PMlinkname{symmetric group}{SymmetricGroup} $S_3$, and $H=\genby{(1,2)}$ and $K=\genby{(1,3)}$, then the two $(H,K)$-double cosets are $\{e,(1,2),(1,3),(1,3,2)\}$ and $\{(2,3),(1,2,3)\}$.