coset (original) (raw)

Let H be a subgroup of a group G, and let a∈G. The left coset of a with respect to H in G is defined to be the set

The right coset of a with respect to H in G is defined to be the set

Two left cosets a⁢H and b⁢H of H in G are either identical or disjoint. Indeed, if c∈a⁢H∩b⁢H, then c=a⁢h1 and c=b⁢h2 for some h1,h2∈H, whence b-1⁢a=h2⁢h1-1∈H. But then, given any a⁢h∈a⁢H, we have a⁢h=(b⁢b-1)⁢a⁢h=b⁢(b-1⁢a)⁢h∈b⁢H, so a⁢H⊂b⁢H, and similarly b⁢H⊂a⁢H. Therefore a⁢H=b⁢H.

Similarly, any two right cosets H⁢a and H⁢b of H in G are either identical or disjoint. Accordingly, the collection of left cosets (or right cosets) partitions the group G; the corresponding equivalence relation for left cosets can be described succintly by the relation a∼b if a-1⁢b∈H, and for right cosets by a∼b if a⁢b-1∈H.

The index of H in G, denoted [G:H], is the cardinality of the set G/H of left cosets of H in G.