Category:Definitions/Normality in Groups

This category contains definitions related to Normality in Groups.
Related results can be found in Category:Normality in Groups.

Let $G$ be a group.

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

$N$ is a normal subgroup of $G$ if and only if:

Definition 1

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

Definition 2

Every right coset of $N$ in $G$ is a left coset

that is:

The right coset space of $N$ in $G$ equals its left coset space.

Definition 3

\(\ds \forall g \in G: \, \)

\(\ds g \circ N \circ g^{-1}\)

\(\subseteq\)

\(\ds N\)

\(\ds \forall g \in G: \, \)

\(\ds g^{-1} \circ N \circ g\)

\(\subseteq\)

\(\ds N\)

Definition 4

\(\ds \forall g \in G: \, \)

\(\ds N\)

\(\subseteq\)

\(\ds g \circ N \circ g^{-1}\)

\(\ds \forall g \in G: \, \)

\(\ds N\)

\(\subseteq\)

\(\ds g^{-1} \circ N \circ g\)

Definition 5

\(\ds \forall g \in G: \, \)

\(\ds N\)

\(=\)

\(\ds g \circ N \circ g^{-1}\)

\(\ds \forall g \in G: \, \)

\(\ds N\)

\(=\)

\(\ds g^{-1} \circ N \circ g\)

Definition 6

\(\ds \forall g \in G: \, \)

\(\ds \leftparen {n \in N}\)

\(\iff\)

\(\ds \rightparen {g \circ n \circ g^{-1} \in N}\)

\(\ds \forall g \in G: \, \)

\(\ds \leftparen {n \in N}\)

\(\iff\)

\(\ds \rightparen {g^{-1} \circ n \circ g \in N}\)

Definition 7

$N$ is a normal subset of $G$.