nLab fixed point group

Definition

Given a group $G$ and an automorphism (or more generally an endomorphism) $\phi \colon G \xrightarrow{\;} G$, its set of fixed points

$$G^\phi \;\coloneqq\; \Big\{ g \in G \;\Big\vert\; \phi(g) = g \Big\} \;\subset\; G$$

evidently constitutes a subgroup, called the fixed-point subgroup of $\phi$.

The analogous statement holds for $G$ replaced by any algebraic structure: The endomorphism property ensures that with a pair of elements being fixed by $\phi$, so is their product $(-)\cdot(-)$ in the algebra:

$$\phi(g_i) = g_i \;\;\;\;\; \Rightarrow \;\;\;\;\; \phi\big( g_1 \cdot g_2 \big) \;=\; \phi(g_1) \cdot \phi(g_2) \;=\; g_1 \cdot g_2$$

Examples

Example

If $\phi = Ad_g$ is an inner automorphism acting by conjugation with an element $g \in G$, then its fixed-point subgroup $G^\phi$ is the centralizer subgroup $C_G(g)$.

Related concepts

• fixed point subspace

References

See also:

• Wikipedia: Fixed-point subgroup