Math and science::Algebra::Aluffi

# Normal subgroup

### Normal subgroup

Let $$G$$ be a group and $$N$$ a subgroup of $$G$$.

$$N$$ is said to be normal iff:

[ $\forall n \in N, \forall g \in G, \; ?$ ]

Or equivalently:

[ $N \text{ is normal } \iff \forall g \in G, \; ?$]

$$gN$$ is syntax for $$\{ a : a = gn \text{ for some } n \in N \}$$, and $$gN = Ng$$ does not imply commutativity.