In group theory
, a subgroup is a subset of a group
that is closed under
the group operation
For example, every permutation group has the subset of even permutations as a subgroup.
A group's largest subgroup is itself; its smallest subgroup is the trivial group consisting of only the identity element.
Every subgroup H of a group G partitions the operations of G into equal sized equivalence classes called cosets; lifted to work on cosets, the group operation forms a group of cosets. Therefore, G can be regarded as a product of
two groups, H and its coset group. If H is a normal divisor of G, G is actually completely determined by them: it is their direct product.