Cohomology and Extensions
For a subgroup, its index is the number of -cosets in .
For a subgroup, its index is the cardinality
of the set of cosets.
If is a subgroup of , the coset projection sends an element of to its orbit .
If is a section of the coset projection , then given by is a bijection. Its inverse is given by the set map given by . Note that the induced product projections conincides with the coset projection.
This argument can be internalized to a group object and a subgroup object in a category . In this case, the coset projection is the coequalizer of the action on by multiplication of . The coset projection need not have a section. However, in case such sections exist, each section of the coset projection, the above argument internalized yields an isomorphism
Even more generally, if is a sequence of subgroup objects, then each section of the projection yields an isomorphism
Returning to the case of ordinary groups, i.e. group objects internal to , where the external axiom of choice is assumed to hold, the coset projection, being a coequalizer and hence an epimorphism, has a section. This gives the multiplicative property of the indices of a sequence of subgroups
The concept of index is meaningful especially for finite groups, i.e. groups internal to . See, for example, its role in the classification of finite simple groups.
Multiplicativity of the index has the following corollary, which is known as Lagrange’s theorem: If is a finite group, then the index of any subgroup is the quotient
of the order (cardinality = number of elements) of by that of .
- For with and the subgroup of the integers given by those that are multiples of , the index is .