---
tag: 0B0WE1GKTK
kind: definition
name: Faithful action
msc: [20B05]
defines: 0P7G9NGG2G
uses: [0YSE802YCQ, 0Z424HK8CQ]
sources:
  - url: https://en.wikipedia.org/wiki/Group_action
    locator: Notable properties of actions
---

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.
The action is faithful when, for each $g \in G$, $g \cdot x = x$ for all $x \in X$ implies $g = e$.

## Remarks

The action is faithful if and only if the associated homomorphism $G \to \operatorname{Sym}(X)$ is injective.
So a faithful action identifies $G$ with a subgroup of $\operatorname{Sym}(X)$.
The action of $G$ on itself by conjugation is faithful if and only if the [[centre|0DZ93493VN]] of $G$ is trivial.

## Explanations

### L1 human

In a faithful action, each element of the group that is not the identity moves at least one point.
So you can tell the group elements apart by watching how they move the points.
The group is then a group of permutations of the set.