Category:Definitions/Normality in Groups
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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 $G$.
Subcategories
This category has the following 7 subcategories, out of 7 total.
N
S
Pages in category "Definitions/Normality in Groups"
The following 21 pages are in this category, out of 21 total.