Information AboutCoset |
| CATEGORIES ABOUT COSET | |
| group theory | |
gH Hg For abelian groups or groups written additively, the notation used changes to ''g''+''H'' and ''H''+''g'' respectively. EXAMPLES The additive cyclic group Z4 = { 0, 1, 2, 3} = ''G'' has a subgroup ''H'' = {0, 2} (isomorphic to Z2). Let us examine the left cosets of ''H'' in ''G''. : 0 + ''H'' = {0, 2} = ''H'' : 1 + ''H'' = {1, 3} : 2 + ''H'' = {2, 0} = ''H'' : 3 + ''H'' = {3, 1} From above, it is clear that there are two distinct cosets, ''H'' itself, and 1+''H'' = 3 + ''H''. Note that ''H'' ∪ 1+''H'' = ''G'', so the different cosets of ''H'' in ''G'' partition ''G''. Since Z4 is an Abelian Group , the right cosets will be the same as the left (this is not difficult to verify). GENERAL PROPERTIES We have ''gH'' = ''H'' if and only if ''g'' is an element of ''H'', since as ''H'' is a subgroup, it must contain the identity. Any two left cosets are either identical or Disjoint -- the left cosets form a Partition of ''G'': every element of ''G'' belongs to one and only one left coset. In particular the identity is only in one coset, and ''H'' itself is the only coset that is a subgroup. We can see this clearly in the above example. The left cosets of ''H'' in ''G'' are the Equivalence Class es under the Equivalence Relation on ''G'' given by ''x'' ~ ''y'' if and only if ''x'' -1''y'' ∈ ''H''. Similar statements are also true for right cosets. A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a Transversal . There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here. Some books on very applied group theory erroneously identify the conjugacy class as 'the' equivalence class as opposed to a particular type of equivalence class. All left cosets and all right cosets have the same number of elements (or allows us to compute the index in the case where G and H are finite, as per the formula: |
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