Re: Looking for a reference

["Zoran Škoda" <zskoda@gmail.com>]

Dear Prof. Porst,

Let us consider the closely related statement (having all the
difficult part from historical point of view and with even closer
proof): If we choose m elements in a group G then there is a
homomorphism from a free group on m generators to G sending the basis
of the free group to these m elements.

This is proved already in

W. Dyck,   Gruppentheoretische Studien. Mathematische Annalen, 20(1),
1–44 (1882) doi:10.1007/bf01443322

In fact, in the statement, he takes the m elements as generators of G
and probably does not comment on uniqueness which is however clear
from the proof. But there is nothing nonobvious to him about passing
to not epimorphic case. He takes the generators of G (that is
epimorphism) just to state a stronger statement telling also about the
kernel, that is equating such G with a factorgroup of the free group
(for which one needs generators, that is epi). According to

Bruce Chandler, Wilhelm Magnus The History of Combinatorial Group
Theory. A Case Study in the History of Ideas. Springer 1982.

further clarifications of Dyck's results in more modern terms of this
and related statements are in

De Séguier, I.-A., 1904, Theorie des Groupes Finis. Elements de la
Theorie des Groupes
Abstraits, 176 pp., Gauthier Villars, Paris.

Magnus's book is also useful as it states Dyck's results in more
modern language than the original.

Now, Dyck is not saying that this is a characterization of the free
group, but regarding that his method studies the kernel of the map the
isomorphism follows. Therefore, the fact has been known to Dyck.
Magnus comments on this on page 10, saying that this is obvious from
the point of view of expositions of de Séguier and of Dehn.

This whole issue in development of group theory and its decisive step
in Dyck's 1882 paper is highly intertwined with the passage from
permutation group theory of earlier times to the abstract group
theory, as studied in detail in the book

Hans Wussing, The genesis of the abstract group concept

which in particular discusses Dyck's paper in Chapter 4.

Now, when it is clear that Dyck's was essentially aware of the
universal property, and stated and proved all needed to make it
obvious, I do not know who first stated it explicitly in full as a
characterization in print. It may be Reidemeister (1926 thesis?),
Nielsen or Otto Schreier.  It should be easier for you to find as most
candidate references are in German.
If it were Dehn, Magnus would probably mention this when discussing
ideas of Dehn's lectures (?) As a property rather than a
characterization of free groups, the universal property has been in
implicit usage through the combinatorial method of group theory much
before 1920s, but not before Dyck.

I hope my reading of Magnus was helpful rather than misleading.

With best regards,
Zoran Škoda

On 7/14/20, porst <porst@uni-bremen.de> wrote:
> Dear all,
>
> If I am not mistaken the characterization of free groups by its universal
> property has first been shown in the late 1920s. Does anybody know a
> reference?
>
> Hans-E. Porst
>
> --
> Hans-E. Porst
> porst@uni-bremen.de <mailto:porst@uni-bremen.de>
>


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