Definition:Normal Subgroup/Definition 2
Definition
Let $G$ be a group.
Let $N$ be a subgroup of $G$.
$N$ is a normal subgroup of $G$ if and only if:
- 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.
Notation
The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \lhd G$.
A normal subgroup is often represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal).
To use the notation introduced in the definition of the conjugate:
- $N \lhd G \iff \forall g \in G: N^g = N$
Also known as
It is usual to describe a normal subgroup of $G$ as normal in $G$.
Some sources refer to a normal subgroup as an invariant subgroup or a self-conjugate subgroup.
This arises from Definition 6:
\(\ds \forall g \in G: \, \) | \(\ds \leftparen {n \in N}\) | \(\iff\) | \(\ds \rightparen {g \circ n \circ g^{-1} \in N}\) | |||||||||||
\(\ds \forall g \in G: \, \) | \(\ds \leftparen {n \in N}\) | \(\iff\) | \(\ds \rightparen {g^{-1} \circ n \circ g \in N}\) |
which is another way of stating that $N$ is normal if and only if $N$ stays the same under all inner automorphisms of $G$.
See Inner Automorphism Maps Subgroup to Itself iff Normal.
Some sources use distinguished subgroup.
Also see
- Results about normal subgroups can be found here.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.6$. Normal subgroups
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): normal subgroup
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): normal subgroup
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): normal subgroup