Difference between revisions of "Coset"
From Online Dictionary of Crystallography
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| − | <font color="blue"> | + | <font color="blue">Co-ensemble</font> (''Fr''). <font color="red">Restklasse</font> (''Ge''). <font color="black">Classe laterale</font> (''It''). <font color="purple">剰余類</font> (''Ja''). |
==Definition== | ==Definition== | ||
| − | If G is a group, H a [[subgroup]] of G, and g an element of G, then | + | If ''G'' is a group, ''H'' a [[subgroup]] of ''G'', and ''g'' an element of ''G'', then |
| − | :gH = { gh : h ∈ H } is a '''left coset of H''' in ''G'' | + | :''gH'' = { ''gh'' : ''h'' ∈ ''H'' } is a '''left coset of ''H'' ''' in ''G'', |
| − | :Hg = { hg : h ∈ H } is a '''right coset of H''' in ''G''. | + | :''Hg'' = { ''hg'' : ''h'' ∈ ''H'' } is a '''right coset of ''H'' ''' in ''G''. |
| − | The decomposition of a group into cosets is unique. Left coset and right cosets however in general do not coincide, unless H is a [[normal subgroup]] of G. | + | The decomposition of a group into cosets is unique. Left coset and right cosets however in general do not coincide, unless ''H'' is a [[normal subgroup]] of ''G''. |
| − | Any two left cosets are either identical or disjoint: the left cosets form a partition of G, because every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and that coset is H itself; this is also the only coset that is a subgroup. The same holds for right cosets. | + | Any two left cosets are either identical or disjoint: the left cosets form a partition of ''G'', because every element of ''G'' belongs to one and only one left coset. In particular the identity is only in one coset, and that coset is ''H'' itself; this is also the only coset that is a subgroup. The same holds for right cosets. |
| − | All left cosets and all right cosets have the same order (number of elements, or cardinality), equal to the order of H, because H is itself a coset. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the '''index''' of H in G, written as [G : H] and given by Lagrange's theorem: | + | All left cosets and all right cosets have the same order (number of elements, or cardinality), equal to the order of ''H'', because ''H'' is itself a coset. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the '''index''' of ''H'' in ''G'', written as [''G'' : ''H''] and given by Lagrange's theorem: |
| − | :|G|/|H| = [G : H]. | + | :|''G''|/|''H''| = [''G'' : ''H'']. |
Cosets are also sometimes called ''associate [[complex]]es''. | Cosets are also sometimes called ''associate [[complex]]es''. | ||
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==See also== | ==See also== | ||
| − | [[Double coset]] | + | *[[Double coset]] |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 10:53, 13 May 2017
Co-ensemble (Fr). Restklasse (Ge). Classe laterale (It). 剰余類 (Ja).
Definition
If G is a group, H a subgroup of G, and g an element of G, then
- gH = { gh : h ∈ H } is a left coset of H in G,
- Hg = { hg : h ∈ H } is a right coset of H in G.
The decomposition of a group into cosets is unique. Left coset and right cosets however in general do not coincide, unless H is a normal subgroup of G.
Any two left cosets are either identical or disjoint: the left cosets form a partition of G, because every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and that coset is H itself; this is also the only coset that is a subgroup. The same holds for right cosets.
All left cosets and all right cosets have the same order (number of elements, or cardinality), equal to the order of H, because H is itself a coset. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H] and given by Lagrange's theorem:
- |G|/|H| = [G : H].
Cosets are also sometimes called associate complexes.
Example
The coset decomposition of the twin lattice point group with respect to the point group of the individual gives the different possible twin laws. Each element in a coset is a possible twin operation.