# Definition:Normal Subset

## Definition

Let $\left({G, \circ}\right)$ be a group.

Let $S \subseteq G$ be a general subset of $G$.

Then $S$ is a normal subset of $G$ if and only if:

### Definition 1

$\forall g \in G: g \circ S = S \circ g$

### Definition 2

$\forall g \in G: g \circ S \circ g^{-1} = S$

or, equivalently:

$\forall g \in G: g^{-1} \circ S \circ g = S$

### Definition 3

$\forall g \in G: g \circ S \circ g^{-1} \subseteq S$

or, equivalently:

$\forall g \in G: g^{-1} \circ S \circ g \subseteq S$

### Definition 4

$\forall g \in G: S \subseteq g \circ S \circ g^{-1}$

or, equivalently:

$\forall g \in G: S \subseteq g^{-1} \circ S \circ g$

### Definition 5

$\forall x, y \in G: x \circ y \in S \implies y \circ x \in S$

### Definition 6

$\map {N_G} S = G$

where $\map {N_G} S$ denotes the normalizer of $S$ in $G$.

### Definition 7

$\forall g \in G: g \circ S \subseteq S \circ g$

or:

$\forall g \in G: S \circ g \subseteq g \circ S$