Subgroup
Definition
If \(H\) is a non empty subset of a group \(G\), and is a group under the same binary operation \(\ast\), then \(H\) is a subgroup of \(G\), often written as \(H \leq G\).
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 \(H \leqslant G\) if and only if \(H\) is a subgroup of \(G\) is a partial order.