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Subgroup

Definition

Given a group $G$, a subgroup is a subset $H \subseteq G$ that is a group.

This means $\forall h_1,h_2 \in H: h_1 h_2 \in H$ and $h_1^{-1} \in H$.

We write $H \leqslant G$.

Criterion

It is sufficient to show that $\forall h_1,h_2 \in H: h_1^{-1}h_2 \in H$.

Properties

For finite groups $H \leqslant G$, Lagrange's Theorem states that $|H|$ divides $|G|$.

Given a collection of subgroups $H_i \leqslant G: i \in I$, their intersection $\bigcap_{i \in I} H_i$ is also a subgroup.

Given a set $S \subseteq G$, the subgroup $\langle S \rangle$ generated by $S$ is the smallest subgroup that contains all of $S$. (The above property means this is well-defined)

Types

A proper subgroup is neither equal to $G$ nor $\{1\}$

A normal subgroup, denoted $H \triangleleft G$, is one where $\forall g \in G: gHg^{-1} = H$ (where $gHg^{-1}=\{ghg^{-1}: h \in H, g \in G\}$). These are essential to defining quotient groups.

all these pages adapted with probably insufficient credit from my university's lecture notes