# Why are faithful actions called faithful and who first called them faithful?

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).

• In French "fidèle" (= faithful) can be used in "fidèle à la réalité", that is to say, which I find translated into "true-to-life" or "true to the reality". I don't know if fidèle was used before or after faithful in the mathematical context, but to me the use of "fidèle" for injective is very natural (here it's injectivity of the map $G\to X^X$— note that faithful is widely used in category theory, for functors, in a similar meaning).
– YCor
May 1 '20 at 23:51
• @YCor In English it can in certain circumstances as well, e.g. in "faithful reproduction". May 1 '20 at 23:52
• @YCor “note that faithful is widely used in category theory, for functors, in a similar meaning” The meaning is not just similar: Faithful group actions are literally a special case of faithful functors. A group G can be regarded as being just a groupoid with a single object. A group action of G on a set X is the same thing as a functor from G to Set sending the single object to X. And the group action is faithful if and only if that functor is faithful in the sense of category theory. May 26 '20 at 20:07
• @YCor is there any doubt that in category theory, the term has been adopted from group action theory? Sep 12 '20 at 6:11
• In the old days "representation" in group theory did not always mean linear representation. So it really meant what later came to be called an action of a group on any sort of object. If we use the word "representation" rather than "action", the point of the word "faithful" becomes clear. The sort of representation that is called faithful gives an accurate picture of the group. Sep 13 '20 at 1:13

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).

• How fitting that this answer was given by someone with a French first name and a German surname!
– RP_
Sep 12 '20 at 16:25
• @Francois thanks a lot. Could you please translate the first German para? (It would be helpful for further viewers and makes your nice answer perfect.) Sep 14 '20 at 12:48
• Sorry, Are you sure it first appeared in Weyl paper? Sep 14 '20 at 15:56
• @C.F.G It translates almost the same as the second: “Since the group structure was abstracted from the representation, the representation is faithful, i.e. to different elements correspond different mappings 𝑈, or, which means the same thing, 𝑈 (𝑠) = 1 only for 𝑠 = 𝟣.” Re: “first” I can’t be positive, but Weyl sounds like he’s introducing the term; earlier papers and books are few (Weyl’s own, Noether, Schur, Speiser, Burnside, Frobenius,...), and I couldn’t find “faithful” in any of them. Maybe you will! Sep 19 '20 at 12:11