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, since the underlying set of is the set of cosets of N in G.

Properties

* If H is a subgroup of G and K is a subgroup of H, then
::
* If H and K are subgroups of G, then
::
:with equality if . (If is finite, then equality holds if and only if .)
* Equivalently, if H and K are subgroups of G, then
::
:with equality if . (If is finite, then equality holds if and only if .)
* If G and H are groups and is a homomorphism, then the index of the kernel

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