Tidbits of terminology and other folklore

Two small and independent interrogative remarks:

(1) Nowadays, an extension of a group G by a group K is a group H fitting in a short exact sequence

1\rightarrow K\rightarrow H\rightarrow G\rightarrow 1

in other words, and rather counterintuitively,  the group G is a quotient of the group which is the extension. When did this terminology originate? A paper of Alan Turing (entitled, rather directly, “The extensions of a group”) defines “extension”, in the very first paragraph, exactly in the opposite (naïve) way, quoting Schreier and Baer who, presumably, had the same convention.

(2) There’s a whole lot of discussion here and there about the mystical “field with one element”; usually, papers of Tits from around 1954 are mentioned as being the source of the whole “idea”; however, the following earlier quote from a 1951 paper  of R. Steinberg (“A geometric approach to the representations of the full linear group over a Galois field”, 1951, p. 279, TAMS 71, 274–282) seems to also contain a germ of the often mentioned analogy between the formulas for the order of the Weyl groups and those of groups of Lie type over a field with q elements, the former being obtained by specializing the latter for q=1:

In closing this section, a remark on the analogy between G and H seems to be in order. Instead of considering G as a group of linear transformations of a vector space, we could consider G as a collineation group of a finite (n-1)-dimensional geometry. If q=1, the vector space fails to exist but the finite geometry does exist and, in fact, reduces to the n vertices of a simplex with a collineation group isomorphic to H. “

In this citation, G is GL(n,Fq), and H is the symmetric group on n letters.

(3) Here’s a third question: when did the terminology “Galois field” become more or less obsolete within the pure mathematics community?

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

5 thoughts on “Tidbits of terminology and other folklore”

  1. Regarding 1., I don’t know the history either, but I will note that in dynamical systems, extensions are defined as they are in group theory: an extension of a dynamical system (X, T) is another system (Y,S) with a surjective map f: Y -> X which intertwines S and T. Perhaps one way to think about this is that the space C(Y) of continuous functions (i.e. observables) of Y extends (contains) the space on X. The opposite notion, that of embedding a dynamical system (X,T) in a superset (Y,S) where Y contains X and T is the restriction of S, is rather boring by comparison, since there is not much coupling going on between X and its complement.

    Anyway, coming from a dynamical perspective, I find it completely natural to view groups as extensions of their quotients (or bundles as extensions of their base spaces, for that matter). As for the other direction, perhaps one should view groups as embedding spaces for their subgroups, rather than extensions…

  2. After a while, it’s true that it also becomes clear that the interesting things that happen in a group extension concern the relation between the quotient and the “big” group, so the emphasis on them is natural.

    Maybe one could speak of covering instead of extensions. Indeed, this is how certain extensions are called (at least in algebraic group theory), and in algebraic geometry, the distinction between (for instance) extensions of fields of functions on a curve and coverings of the curves is maintained on the terminological level, despite the complete equivalence between the two.

  3. Regarding (2), the attribution of the idea of a “field with one element” to Tits seems justified for me. This is despite the fact that it was probably known to 19th century enumerative algebraic geometers that there is a relation between the cell structure of flag varieties and reflection groups. So while I believe that the phenomenon mentioned by Steinberg was observed long before, it was then only a numerical curiosity. On the other hand, Tits really makes the concept of a geometry over a field with one element a tool that is used to prove something, which is something completely different. Of course, he says apartment rather than “geometry over a field with one element”, but that is just a question of terminology. So, I think, the IDEA to use the field with one element is rightfully attributed to Tits. As far as the TERM “field with one element” is concerned, I would be curious to learn, who introduced it. It does not seem to appear in the early work of Tits nor in the work of Steinberg quoted above. Any references?

  4. The group Ext^1(A,B) has to be written that way because Hom(A,B)=Ext^0(A,B) has to be written that way. (Ext^1 is the first derived functor of Hom.) But then Ext^1(A,B) classifies extensions of A by B. If you used the other terminology, then the order wouldn’t agree with the order of the symbols on the page.

    I have no idea if that’s the historical reason, but I’m very happy someone decided to do it this way. Otherwise I’d go crazy.

  5. That’s a good point.
    On the other hand, many people have gone crazy with the fact that composition of maps is almost alway written in the opposite order that the arrows appear on paper when drawing diagrams…

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