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| − | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266201.png" /> by a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266202.png" /> (from the left)'' | + | ''$G$ by a subgroup $H$ (from the left)'' |
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| − | A set of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266203.png" /> of the form | + | A set of elements of $G$ of the form |
| | + | $$ |
| | + | aH = \{ ah : h \in H \} |
| | + | $$ |
| | + | where $a$ is some fixed element of $G$. This coset is also called the left coset by $H$ in $G$ defined by $a$. Every left coset is determined by any of its elements. $aH = H$ if and only if $a \in H$. For all $a,b \in G$ the cosets $aH$ and $bH$ are either equal or disjoint. Thus, $G$ decomposes into pairwise disjoint left cosets by $H$; this decomposition is called the left decomposition of $G$ with respect to $H$. Similarly one defines right cosets (as sets $Ha$, $a \in G$) and also the right decomposition of $G$ with respect to $H$. These decompositions consist of the same number of cosets (in the infinite case, their cardinalities are equal). This number (cardinality) is called the index of the subgroup $H$ in $G$. For normal subgroups, the left and right decompositions coincide, and in this case one simply speaks of the decomposition of a group with respect to a normal subgroup. |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266204.png" /></td> </tr></table>
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266205.png" /> is some fixed element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266206.png" />. This coset is also called the left coset by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266207.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266208.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c0266209.png" />. Every left coset is determined by any of its elements. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662010.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662011.png" />. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662012.png" /> the cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662014.png" /> are either equal or disjoint. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662015.png" /> decomposes into pairwise disjoint left cosets by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662016.png" />; this decomposition is called the left decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662017.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662018.png" />. Similarly one defines right cosets (as sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662020.png" />) and also the right decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662021.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662022.png" />. These decompositions consist of the same number of cosets (in the infinite case, their cardinalities are equal). This number (cardinality) is called the index of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026620/c02662024.png" />. For normal subgroups, the left and right decompositions coincide, and in this case one simply speaks of the decomposition of a group with respect to a normal subgroup.
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| | + | ====Comments==== |
| | + | See also [[Normal subgroup]]. |
| | | | |
| | + | The (left) cosets are the equivalence classes for the [[equivalence relation]] $a \sim b \Leftrightarrow a^{-1}b \in H$. |
| | | | |
| − | ====Comments====
| + | [[Category:Group theory and generalizations]] |
| − | See also [[Normal subgroup|Normal subgroup]].
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Latest revision as of 20:26, 4 December 2014
$G$ by a subgroup $H$ (from the left)
A set of elements of $G$ of the form
$$
aH = \{ ah : h \in H \}
$$
where $a$ is some fixed element of $G$. This coset is also called the left coset by $H$ in $G$ defined by $a$. Every left coset is determined by any of its elements. $aH = H$ if and only if $a \in H$. For all $a,b \in G$ the cosets $aH$ and $bH$ are either equal or disjoint. Thus, $G$ decomposes into pairwise disjoint left cosets by $H$; this decomposition is called the left decomposition of $G$ with respect to $H$. Similarly one defines right cosets (as sets $Ha$, $a \in G$) and also the right decomposition of $G$ with respect to $H$. These decompositions consist of the same number of cosets (in the infinite case, their cardinalities are equal). This number (cardinality) is called the index of the subgroup $H$ in $G$. For normal subgroups, the left and right decompositions coincide, and in this case one simply speaks of the decomposition of a group with respect to a normal subgroup.
See also Normal subgroup.
The (left) cosets are the equivalence classes for the equivalence relation $a \sim b \Leftrightarrow a^{-1}b \in H$.
How to Cite This Entry:
Coset in a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coset_in_a_group&oldid=16601
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article