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'double coset'
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| Title of object: |
double coset |
| Canonical Name: |
DoubleCoset |
| Type: |
Definition |
| Created on: |
2006-10-01 11:26:31 |
| Modified on: |
2006-10-01 11:26:31 |
| Classification: |
msc:20A05 |
Revision comment (for changes between this and next version):
Preamble:
| \def\genby#1{\langle#1\rangle} |
Content:
\PMlinkescapeword{coset}
\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 every $(H,K)$-double coset is a union of right cosets of $H$,
and also a union of left cosets of $K$.
The $(H,K)$-double cosets form 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 $\{(2,3),(1,2,3)\}$.
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