Sorry for this question. I asked this on [MSE](https://math.stackexchange.com/q/3647129/272127) and [HSM](https://hsm.stackexchange.com/questions/11724/why-is-faithful-actions-called-faithful-and-who-first-called-it-faithful) but no one answered and I decided to post it here that is full of experts.




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I want to know *why* are faithful actions called faithful and *who* first called them faithful?


**Definition:** An action $G$ on $X$ is faithful when ${g\_1 \neq g\_2 \Rightarrow g\_1 x \neq g\_2 x}$ for some ${x \in X}$ (different elements of $G$ act differently at some point).



Why are faithful actions called faithful and who first called them faithful?
The German word is *treu*, and I would look to papers by Hermann Weyl for its introduction. E.g. *Quantenmechanik und Gruppentheorie* ([1927](//zbmath.org/?q=an:53.0848.02), p. [16](//doi.org/10.1007/BF02055756)):



> 
> Da das Gruppenschema aus der Darstellung abstrahiert wurde, ist die Darstellung ***getreu***, d.h. verschiedenen Elementen entsprechen verschiedene Abbildungen $U$, oder, was dasselbe besagt, $U(s)$ ist $= \mathbf1$ nur für $s = \mathsf1$.
> 
> 
> 


or *The theory of groups and quantum mechanics* ([1931](//ams.org/mathscinet-getitem?mr=3363447), p. [114](//archive.org/details/quantumtheoryofa029235mbp/page/n139) — note the scare quotes):



> 
> The realization is said to be ***faithful***
> when to distinct elements of the group correspond distinct
> transformations:
> $$
> T(a)\ne T(b)\text{ when } a\ne b.
> $$
> In accordance with the fundamental equation (2.1) the necessary
> and sufficient condition for “faithfulness” is that $T(a)$ shall be
> the identity *only if* $a$ is the unit element.
> 
> 
> 


The same definition occurs in Wigner ([1931](//zbmath.org/?q=an:57.1578.03), p. [79](//doi.org/10.1007/978-3-663-02555-9)), van der Waerden ([1931](//zbmath.org/?q=an:57.0153.03), p. [178](//doi.org/10.1007/978-3-662-42016-4); [1932](//zbmath.org/?q=an:58.0121.03), p. [32](//doi.org/10.1007/978-3-662-02187-3)), and in French, Bauer ([1933](//zbmath.org/?q=an:59.1521.02), p. [75](http://www.numdam.org/item/AIHP_1933__4_1_1_0)).


