Subgroups#

Definition: Subgroup

Let \((G, \cdot)\) be a group and let \(S \subseteq G\).

We say that \(S\) is a subgroup of \((G, \cdot)\) if \(S\) is itself a group \((S, \cdot)\) under the operation \(\cdot\).

Example: Trivial Subgroups

Every group is a subgroup of itself.

If \(e\) is the identity element of a group, then \(\{e\}\) is a subgroup of it.

Example

If \(k \in \mathbb{N}_0\), then \(\{kx \mid x \in \mathbb{Z}\}\) is a subgroup of \((\mathbb{Z}, +)\).

Example

If \(a, b \in \mathbb{R}\), then \(\{(xa, xb) \in \mathbb{R}^2 \mid x \in \mathbb{R}\}\) is a subgroup of \((\mathbb{R}^2, +)\)

Theorem: Equivalent Definition

Let \((G, \cdot)\) be a group and let \(S \subseteq G\).

Then \((S, \cdot)\) is a subgroup of \((G, \cdot)\) if and only if

\[ a, b \in S \implies a^{-1} \in S \qquad \text{and} \qquad ab \in S \]

Proof

TODO