Characterizations and Examples of Normal Subgroups

In the setting of Groups: Standing Notation, let $G$ be a group and $N$ a subgroup of $G$.

(i) Normality by cosets. $N$ is normal if and only if its left and right cosets agree:

$$gN=Ng\qquad\text{for every } g\in G.$$

(ii) Normality by conjugates. $N$ is normal if and only if it equals each of its conjugates:

$$\{gng^{-1}:n\in N\}=N\qquad\text{for every } g\in G.$$

(iii) Abelian groups. If $G$ is abelian, then $N$ is normal.

(iv) Trivial subgroups. The subgroups $\{e\}$ and $G$ of $G$ (Subgroup Criterion and Basic Examples (iv)) are normal.