This preview shows page 1. Sign up to view the full content.
Unformatted text preview: H in G . Proof. The distinct left cosets of H are mutually disjoint by Proposition 2.5.4 and each has the same size (namely j H j D j eH j ) by Proposition 2.5.5 . Since the union of the left cosets is G , the cardinality of G is the cardinality of H times the number of distinct left cosets of H . n Deﬁnition 2.5.7. For a subgroup H of a group G , the index of H in G is the number of left cosets of H in G . The index is denoted ŒG W HŁ . Index also makes sense for inﬁnite groups. For example, take the larger group to be Z and the subgroup to be n Z . Then Œ Z W n Z Ł D n , because there are n cosets of n Z in Z . Every subgroup of Z (other...
View Full Document
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08