7. Subgroups

A subgroup is a group that sits inside a larger group, using the same operation

Let $G$ be a group and $H \subseteq G$

Then $H$ is a subgroup of $G$ if $H$ is itself a group under the same operation

Key Points

Associativity is automatic if $\cdot$ is associative in $G$

The other Composition Rules must hold within $H$ for $H$ to be considered a group

Any subset of a group that satisfies closure, contains the identity, and contains inverses is a subgroup

Example 1

Integers ($\Z$)

$\Z$ is a subgroup of $\mathbb{Q}, \mathbb{R}, \mathbb{C}$ under addition

Example 2

Circle Group

$$S^1 = \{z \in \mathbb{C} : |z| = 1\}$$

is a subgroup of the non-zero complex numbers $\mathbb{C}^*$ under multiplication

Example 3

Rotations of a Triangle

$$\{Id, r, r^2\}$$

the rotations in $D_3$, form a subgroup of the dihedral group of the triangle