Subgroup

Definition

If H is a non empty subset of a group G, and is a group under the same binary operation ∗, then H is a subgroup of G, often written as H≤G.

A subset of a group which is a group need not be a subgroup, specifically because it must be a group under the same operation.

To check if a subset of a group is a subgroup, not all axioms need to be checked, see subgroup tests.

Partial Order

Theorem

The subgroup relation H⩽G if and only if H is a subgroup of G is a partial order.