Definition:Normal Subgroup/Also known as
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Definition
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.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.10$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64$