Cohomology and Extensions
For a subgroup inclusion, its index is the number of -cosets in , hence roughly is the number of copies of that appear 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 FinSet. 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 .