Subgroups

Introduction

Within a group, there often exist smaller groups. A subgroup is a subset of a group that itself forms a group under the same operation.

Definition

A subgroup $H$ of a group $G$ is a non-empty subset of $G$ that itself forms a group under the binary operation of $G$ . This is denoted by $H \leq G$ .

Types of Subgroups

If $H$ is a proper subset of $G$ ( $H \neq G$ ), it is a proper subgroup, denoted $H < G$
The subgroup ${e}$ consisting of only the identity element is the trivial subgroup
The group $G$ itself is always a subgroup of $G$

Subgroup Tests

To verify that a non-empty subset $H \subseteq G$ is a subgroup, one can use the following tests:

Two-Step Test

$H$ is a subgroup if and only if:

It is closed under the group operation (for all $a, b \in H$ , $a b \in H$ )
It is closed under inverses (for all $a \in H$ , $a^{- 1} \in H$ )

One-Step Test

$H$ is a subgroup if and only if for all $a, b \in H$ , the element $a b^{- 1}$ is also in $H$ .

Finite Subgroup Test

If $H$ is a finite, non-empty subset of $G$ , it is a subgroup if and only if it is closed under the group operation.

Examples

Example 1: Subgroups of $(Z, +)$

The set of even integers $2 Z = {0, \pm 2, \pm 4, \dots}$ is a subgroup
For any $n \in N$ , the set $n Z = {0, \pm n, \pm 2 n, \dots}$ is a subgroup
The set ${0}$ is the trivial subgroup

Example 2: Subgroups of $D_{4}$

The set of rotations ${e, r, r^{2}, r^{3}}$ is a subgroup
The set ${e, r^{2}}$ is a subgroup (rotations by $0 °$ and $180 °$ )
The set ${e, s}$ is a subgroup (identity and horizontal reflection)
The set ${e, s r^{2}}$ is a subgroup (identity and diagonal reflection)

Example 3: Subgroups of $S_{3}$

The alternating group $A_{3} = {e, (1, 2, 3), (1, 3, 2)}$ is a subgroup
Any cyclic subgroup generated by a single element, e.g., $⟨ (1, 2) ⟩ = {e, (1, 2)}$
The trivial subgroup ${e}$

Example 4: Subgroups of $(R^{*}, \cdot)$

The positive real numbers $R^{+}$ form a subgroup
The set ${1, - 1}$ forms a subgroup
The set ${1}$ is the trivial subgroup

Properties of Subgroups

Intersection of Subgroups

The intersection of any collection of subgroups of $G$ is itself a subgroup of $G$ .

Subgroup Generated by a Set

For any subset $S \subseteq G$ , the subgroup generated by $S$ , denoted $⟨ S ⟩$ , is the smallest subgroup of $G$ containing $S$ .

Cyclic Subgroups

If $S = {a}$ is a singleton, then $⟨ a ⟩ = {a^{n} ∣ n \in Z}$ is called the cyclic subgroup generated by $a$ .

Lagrange's Theorem

Theorem: If $G$ is a finite group and $H$ is a subgroup of $G$ , then the order of $H$ divides the order of $G$ .

This theorem places a strong restriction on the possible sizes of subgroups.

Examples of Subgroup Calculations

Example 1: Finding All Subgroups of $Z_{12}$

The subgroups of $Z_{12}$ are:

${0}$ (order 1)
${0, 6}$ (order 2)
${0, 4, 8}$ (order 3)
${0, 3, 6, 9}$ (order 4)
${0, 2, 4, 6, 8, 10}$ (order 6)
$Z_{12}$ itself (order 12)

Example 2: Subgroups of $D_{6}$

The dihedral group $D_{6}$ has order 12. Its subgroups include:

${e}$ (order 1)
${e, r^{3}}$ (order 2)
${e, r^{2}, r^{4}}$ (order 3)
${e, r^{3}, s, s r^{3}}$ (order 4)
${e, r, r^{2}, r^{3}, r^{4}, r^{5}}$ (order 6)
$D_{6}$ itself (order 12)