A subgroup of a group \((G,*)\) is a group of the form \((H,*)\), where \(H \subset G\). We usually say simply that \(H\) is a subgroup of \(G\).

For a subset of a group \(G\) to be a subgroup, it needs to satisfy all of the group axioms itself: closure, associativity, identity, and inverse. We get associativity for free because \(G\) is a group. So the requirements of a subgroup \(H\) are:

1. Closure: For any \(x, y\) in \(H\), \(x*y\) is in \(H\).

2. Identity: The identity \(e\) of \(G\) is in \(H\).

3. Inverses: For any \(x\) in \(H\), \(x^{-1}\) is also in \(H\).

A subgroup is called normal if it is closed under conjugation.

The subgroup relation is transitive: if \(H\) is a subgroup of \(G\), and \(I\) is a subgroup of \(H\), then \(I\) is a subgroup of \(G\).

Examples

Any group is a subgroup of itself. The trivial group is a subgroup of every group.

For any integer \(n\), the set of multiples of \(n\) is a subgroup of the integers (under addition).