The right coset of with respect to in is defined to be the set
Two left cosets and of in are either identical or disjoint. Indeed, if , then and for some , whence . But then, given any , we have , so , and similarly . Therefore .
Similarly, any two right cosets and of in are either identical or disjoint. Accordingly, the collection of left cosets (or right cosets) partitions the group ; the corresponding equivalence relation for left cosets can be described succintly by the relation if , and for right cosets by if .
The index of in , denoted , is the cardinality of the set of left cosets of in .
|Date of creation||2013-03-22 12:08:10|
|Last modified on||2013-03-22 12:08:10|
|Last modified by||djao (24)|