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Cosets

For G a group and H a subgroup, a right coset of H in G is

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Similarly we can define a left coset

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Typically, a left coset gH and a right coset Hg are not equal as sets. In additive notation, a right coset would be denoted as H+g and a left coset as g+H.

The right (left) cosets of a subgroup have the following properties

Similar properties hold for left cosets. We say that the cosets partition the group G. Indeed, they are the equivalence classes of the equivalence relation on G defined by tex2html_wrap_inline1301 (mod H) if and only if tex2html_wrap_inline1289 .

For example, the group tex2html_wrap_inline1307 consists of the 12 even permutations of tex2html_wrap_inline1257 . The set tex2html_wrap_inline1311 is a subgroup. The right cosets of H in tex2html_wrap_inline1307 are:

The left cosets of H in tex2html_wrap_inline1307 are: Note that the number of elements in each coset is the same and that is not accidental. Also note that, other than the subgroup H, no left coset is a right coset.

The map tex2html_wrap_inline1339 defined by tex2html_wrap_inline1341 is a bijection. In particular for G a finite group, all cosets have the same number of elements. This is the basis for a proof of Lagrange's Theorem.

The number of cosets of H in G (right or left) is called the index and is denoted by tex2html_wrap_inline1349 . For G finite, we have tex2html_wrap_inline1353 . Thus the index multiplied by the order of the subgroup is the order of the group. Thus, for G finite, both the index and the order of the subgroup divide the order of G.

A right transversal for H in G is a set consisting of one element from each right coset. This is the same as a set of equivalence class representatives. A left transversal is defined similarly.




next up previous
Next: Normal subgroups Up: Groups Previous: Generating sets

Peter Williams
Sun Mar 30 14:48:35 PST 1997