Definition:Normalizer

Definition

Let $G$ be a group.

Let $S$ be a subset of $G$.

Then the normalizer of $S$ in $G$ is the set $\map {N_G} S$ defined as:

$\map {N_G} S := \set {a \in G: S^a = S}$

where $S^a$ is the $G$-conjugate of $S$ by $a$.

If $S$ is a singleton such that $S = \set s$, we may also write $\map {N_G} s$ for $\map {N_G} S = \map {N_G} {\set s}$, as long as there is no possibility of confusion.

Also denoted as

The notation $\map N {S; G}$ is sometimes seen for the normalizer of $S$ in $G$.

Also see

• Normalizer is Subgroup

• Results about normalizers can be found here.

Linguistic Note

The UK English spelling of normalizer is normaliser.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.6$. Stabilizers: Example $110$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Exercise $25.20$
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Exercise $5.15$
• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35 \gamma$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 48$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $12$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$