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 $N_G \left({S}\right)$ such that:

$N_G \left({S}\right) = \left\{{a \in G: S^a = S}\right\}$

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

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

The UK English spelling of this is normaliser.

Also see

• Normalizer is Subgroup

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $25.20$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.9$: Exercise $5.15$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 35 \gamma$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 48$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $8.12$