Subgroup

Equivalently, a subset H of G is a subgroup of G if and only if:

• (a) e_G in H;
• (b) x,y in H ==> xy in H;
• (c) x in H ==> x^-1 in H.

For example the even integers form a subgroup of the additive group of the integers (Z,+).

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.

Sub"group` (?), n. Biol.

A subdivision of a group, as of animals.

Darwin.

© Webster 1913.