TITLE: Why are faithful actions called faithful and who first called them faithful?
QUESTION [9 upvotes]: Sorry for this question. I asked this on MSE and HSM but no one answered and I decided to post it here that is full of experts.

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).

REPLY [16 votes]: The German word is treu, and I would look to papers by Hermann Weyl for its introduction. E.g. Quantenmechanik und Gruppentheorie (1927, p. 16):

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, p. 114 — 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, p. 79), van der Waerden (1931, p. 178; 1932, p. 32), and in French, Bauer (1933, p. 75).