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For a subgroup inclusion, its index is the number of -cosets in , hence roughly is the number of copies of that appear in .
If is a subgroup of , the coset coprojection sends an element of to its orbit .
If is a section of the coset coprojection , then given by is a bijection. Its inverse is given by the set map given by . To note that the induced product coprojections coincides with the coset coprojection.
This argument can be internalized to appy to group objects with subgroup objects in a suitable category . In this generality, the coset coprojection is the coequalizer of the action on by multiplication of .
The coset coprojection need not have a section. However, in case such sections exist, each section of the coset coprojection, 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 Set, where the external axiom of choice is assumed to hold, the coset coprojection, 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 .
Last revised on April 30, 2025 at 10:53:30. See the history of this page for a list of all contributions to it.