Definition:Stabilizer

Theorem

Let $G$ be a group which acts on a set $X$.

For each $x \in X$, the stabilizer of $x$ by $G$ is defined as:

$\operatorname{Stab} \left({x}\right) := \left\{{g \in G: g * x = x}\right\}$

where $*$ denotes the group action.

Alternative Notation

Some authors use $G_x$ for the stabilizer of $x$ by $G$.

Linguistic Note

The English spelling for it is stabiliser.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.6$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 54$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $6.5$
• John F. Humphreys: A Course in Group Theory (1996): $\S 10$: Definition $10.8$