Looking for a reference

Looking for a reference

[porst]

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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Re: Looking for a reference

[Johannes Huebschmann]

This is implicit in Ch. 2 of

@book{0004.36904,
author="Reidemeister, Kurt",
title="{Einf\"uhrung in die kombinatorische Topologie.}",
language="German",
publisher="{Braunschweig: Friedr. Vieweg \& Sohn A.-G. XII, 209 S. }",
year="1932",
keywords="{topology}",
}
Every group is a homomorphic image of a free group.

Presumably it also implicit in

@Article{zbMATH02581122,
  Author = {O. {Schreier}},
  Title = {{Die Untergruppen der freien Gruppen.}},
  FJournal = {{Abhandlungen aus dem Mathematischen Seminar der Universit\"at Hamburg}},
  Journal = {{Abh. Math. Semin. Univ. Hamb.}},
  ISSN = {0025-5858; 1865-8784/e},
  Volume = {5},
  Pages = {161--183},
  Year = {1927},
  Publisher = {Springer, Berlin/Heidelberg},
  Language = {German},
  Zbl = {53.0110.01}
}



Johannes



HUEBSCHMANN Johannes
Professeur émérite
Université de Lille - Sciences et Technologies 
Département de Mathématiques
CNRS-UMR 8524 Laboratoire Paul Painlevé
Labex CEMPI (ANR-11-LABX-0007-01)
59 655 VILLENEUVE D'ASCQ Cedex/France
http://math.univ-lille1.fr/~huebschm

Johannes.Huebschmann@univ-lille.fr





----- Mail original -----
De: "porst" <porst@uni-bremen.de>
À: "categories@mta.ca list" <categories@mta.ca>
Envoyé: Mardi 14 Juillet 2020 18:08:51
Objet: categories: Looking for a reference

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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Re: Looking for a reference

[Zoran Škoda]

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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Re: looking for a reference...

[Fred E.J. Linton]

Referring to Alex's question in Dana's original post (below):

Sorry, but I'm not seeing how this is all that different from just
picking an object A in C, contemplating all powers of A in C, and
asking about the full subcategory K (of C) of all subobjects 
of those powers?

In case C is a variety, surely that's a well-understood sort of class
of algebras (closed under products and subalgebras), n'est-ce pas?

Cheers, -- Fred

---

------ Original Message ------
Received: Sun, 14 Sep 2014 09:28:51 AM EDT
From: Dana Scott <dana.scott@cs.cmu.edu>
To: Categories list <categories@mta.ca>
Cc: Alex Kruckman <kruckman@gmail.com> 
Subject: categories: Re: looking for a reference...

> If you have comments/suggestions, please reply to Mr. Kruckman.  Thanks.
> 
> On Sep 13, 2014, at 10:02 AM, Alex Kruckman <kruckman@gmail.com> wrote:
> 
>> Professor Scott,
>> 
>> In writing up some work I did with another graduate student, we’ve
>> noticed that one argument is really a special case of a very general
>> fact. It's easy to prove, and it's quite nice, but I've never seen it
>> explicitly noted. Have you?
>> 
>> Here it is:
>> 
>> 1. Suppose we have a contravariant functor F from Sets to some other
>> category C which turns coproducts into products. This functor
automatically
>> has an adjoint, given by G(-) = Hom_C(-,F(1)), where 1 is the one element
>> set. If you like, the existence of G is an instance of the special
adjoint
>> functor theorem, but it's also easy to check by hand. The key thing is
that
>> every set X can be expressed as the X-indexed coproduct of copies of the
one
>> element set, so we have (the = signs here are natural isomorphisms):
>> 
>> Hom_C(A,F(X)) = Hom_C(A,F(coprod_X 1)) = Hom_C(A,prod_X F(1)) =
>> 	prod_X Hom_C(A,F(1)) = prod_X G(A) = Hom_Set(X,G(A))
>> 
>> 2. Now let's say the category C is the category of algebras in some
signature.
>> Let's call algebras in the image of F "full", and let's say we're
interested
>> in the class K of subalgebras of full algebras. This class is closed
under
>> products and subalgebras, so if it's elementary, then it has an
axiomatization
>> by universal Horn sentences (i.e. it's a quasivariety), and moreover every
algebra
>> in the class is a subalgebra of a product of copies of F(1), so a
universal Horn
>> sentence is true of every algebra in the class if and only if it's true of
F(1).
>> 
>> 3. Okay, let's say we have an axiomatization T for K. Then we have a
“representation
>> problem": given an algebra A satisfying T, embed it in some full algebra.
Well,
>> there's a canonical such embedding, given by the unit of the adjunction A
-> F(G(A)).
>> That is, A -> F(Hom_C(A,F(1))). 
>> 
>> Examples of these observations include all the constructions of algebras
from
>> sets by powerset - the Stone representation theorem for Boolean algebras
(minus the
>> topology, of course), but also the representation theorems for lattices,
semilattices, etc.
>> 
>> Thanks for taking the time to read this. Let me know if it rings a bell.
>> 
>> -Alex
> 



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Re: looking for a reference...

[Dana Scott]

If you have comments/suggestions, please reply to Mr. Kruckman.  Thanks.

On Sep 13, 2014, at 10:02 AM, Alex Kruckman <kruckman@gmail.com> wrote:

> Professor Scott,
> 
> In writing up some work I did with another graduate student, we’ve
> noticed that one argument is really a special case of a very general
> fact. It's easy to prove, and it's quite nice, but I've never seen it
> explicitly noted. Have you?
> 
> Here it is:
> 
> 1. Suppose we have a contravariant functor F from Sets to some other
> category C which turns coproducts into products. This functor automatically
> has an adjoint, given by G(-) = Hom_C(-,F(1)), where 1 is the one element
> set. If you like, the existence of G is an instance of the special adjoint
> functor theorem, but it's also easy to check by hand. The key thing is that
> every set X can be expressed as the X-indexed coproduct of copies of the one
> element set, so we have (the = signs here are natural isomorphisms):
> 
> Hom_C(A,F(X)) = Hom_C(A,F(coprod_X 1)) = Hom_C(A,prod_X F(1)) =
> 	prod_X Hom_C(A,F(1)) = prod_X G(A) = Hom_Set(X,G(A))
> 
> 2. Now let's say the category C is the category of algebras in some signature.
> Let's call algebras in the image of F "full", and let's say we're interested
> in the class K of subalgebras of full algebras. This class is closed under
> products and subalgebras, so if it's elementary, then it has an axiomatization
> by universal Horn sentences (i.e. it's a quasivariety), and moreover every algebra
> in the class is a subalgebra of a product of copies of F(1), so a universal Horn
> sentence is true of every algebra in the class if and only if it's true of F(1).
> 
> 3. Okay, let's say we have an axiomatization T for K. Then we have a “representation
> problem": given an algebra A satisfying T, embed it in some full algebra. Well,
> there's a canonical such embedding, given by the unit of the adjunction A -> F(G(A)).
> That is, A -> F(Hom_C(A,F(1))). 
> 
> Examples of these observations include all the constructions of algebras from
> sets by powerset - the Stone representation theorem for Boolean algebras (minus the
> topology, of course), but also the representation theorems for lattices, semilattices, etc.
> 
> Thanks for taking the time to read this. Let me know if it rings a bell.
> 
> -Alex



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