Normal subgroup

A subgroup $N\le G$ is normal when $gN{g}^{-1}=N$ for every $g\in G$.

A minimal normal subgroup is a nontrivial normal subgroup containing no smaller nontrivial normal subgroup of the ambient group. It need not be a minimal nontrivial subgroup. In a finite soluble group, it is elementary abelian: its derived subgroup must be trivial, a characteristic Sylow subgroup selects one prime, and the characteristic subgroup of elements killed by that prime gives exponent $p$.

If $N=\{1,z\}$ is a normal subgroup of a group $G$, conjugation by any element of $G$ preserves $N$ and fixes $1$. It must therefore fix $z$ as well. Thus $N$ lies in the center of a group. This fact can turn a small normal subgroup into a commuting factor without assuming the whole group is abelian.

The core of a subgroup $H\le G$ isIt is the largest normal subgroup of $G$ contained in $H$. If $H$ has finite index $k$, the action on $G/H$ gives $[G:{\mathrm{Core}}_G(H)]\le k!$.

The normal closure of a subset $S\subseteq G$ is the smallest normal subgroup containing $S$, equivalently the subgroup generated by all conjugates $gs{g}^{-1}$ with $g\in G$ and $s\in S$.

The preimage of a normal subgroup under any group homomorphism is normal.

Under a surjective homomorphism, the image of a normal subgroup is normal.