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Normal subgroup

Definition

Given a subgroup $N \leqslant G$, if $\forall g \in G: gNg^{-1} = N$, then we say $N$ is a normal subgroup and write $N \triangleleft G$.

Equivalently, for all $g \in G$, the left-coset $gN$ and right-coset $Ng$ are equal

Properties

If $H \triangleleft N \triangleleft G$, we don't necessarily have $H \triangleleft G$. TODO compare with "characteristic subgroup".

Given $N \triangleleft G$, we can define the quotient group $G/N$ as the set of cosets $\{gN: g \in N\}$ with the operation $g_1 N . g_2 N = (g_1.g_2) N$.

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

Given a set $S \subseteq G$, the normal closure $\langle\langle S \rangle\rangle$ is the smallest normal subgroup that contains all of $S$ (the above property means this is well-defined). This is given by $\langle\langle S \rangle\rangle = \langle gsg^{-1}: s \in S, g \in G \rangle$.

Given $H \leqslant G$ and $N \triangleleft G$, we have $H \cap N$ is a normal subgroup of $H$ (but not necessarily of $G$). Special case: if $N \leqslant H \leqslant G$ and $N \triangleleft G$, then $N \triangleleft H$

Given $N \triangleleft G$ and $H \leqslant G$, the set $HN = \{hn: h \in H; n \in N\}$ is a subgroup of $G$ and $N \triangleleft HN$. See second isomorphism theorem for further properties.

Types

TODO characteristic subgroup

Examples

The commutator subgroup is the group generated by the set of commutators $\{ghg^{-1}h^{-1}: g,h \in G\}$; this is a normal subgroup; in fact, a characteristic subgroup.

Related theorems

TODO simple group

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