Subgroups

Principle

A subgroup is a subset of a group that is itself a group using the same operation. The operation is not reinvented; it is inherited from the parent group.

Subgroups let a large symmetry system be studied through smaller collections of transformations, such as only the rotations inside a full polygon symmetry group.

Notation

• \((G,*)\) is the parent group.
• \(H\) is a subset of \(G\).
• \(H\subseteq G\) means every element of \(H\) is also an element of \(G\).
• \(e\) is the identity element of the parent group.
• \(h\) is an element of \(H\).
• \(h^{-1}\) is the inverse of \(h\) in the parent group.
• \(H\le G\) means \(H\) is a subgroup of \(G\).

Method

To test whether \(H\) is a subgroup of \(G\):

1. Check that \(H\) is not empty.
2. Check that if \(h_1,h_2\in H\), then \(h_1*h_2\in H\).
3. Check that if \(h\in H\), then \(h^{-1}\in H\).
4. Use the same operation \(*\) as the parent group.

Associativity does not need a separate proof once \(H\) uses the operation from \(G\), because associativity is inherited from the parent group.

Rules

• Every subgroup contains the identity element of the parent group.
• A subgroup uses the same operation as the parent group.
• Closure under products keeps the subset stable under the operation.
• Closure under inverses makes every allowed operation reversible inside the subset.
• A subset that omits required inverses is not a subgroup.

Examples

Let \(2\mathbb Z\) be the set of even integers. Test it as a subgroup of \((\mathbb Z,+)\).

The set is non-empty because \(0\in2\mathbb Z\).

For closure under addition, write two even integers as \(2m\) and \(2n\), where \(m,n\in\mathbb Z\). Then:

For inverses under addition, the inverse of \(2m\) is \(-(2m)\). Since \(-m\) is also an integer:

The identity is \(0\), and \(0=2\cdot0\), so \(0\in2\mathbb Z\). Therefore \(2\mathbb Z\le\mathbb Z\) under addition.

Checks

• Use the same operation as the parent group.
• Check that inverses stay inside the subset.
• Check that the identity is the parent identity, not a new element.
• Do not assume every subset of a group is a subgroup.