Category:Definitions/Normal Subgroups

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This category contains definitions related to Normal Subgroups.
Related results can be found in Category:Normal Subgroups.


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 $G$.

Subcategories

This category has the following 2 subcategories, out of 2 total.

G

M

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