%%% collection of examples est' motivacia
\documentclass[11pt,english]{smfart}%\documentclass[a4paper,10pt]{article}
%\documentclass{demorgan}
\usepackage[T2A]{fontenc}
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\usepackage[english,russian]{babel}
\usepackage{url}
\usepackage{smfthm}
\usepackage{hyperref}
\usepackage{MnSymbol}
\usepackage{graphicx}
\usepackage[matrix, arrow,all,cmtip,color]{xy}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
\def\lra{\longrightarrow}
\def\ra{\rightarrow}
\def\rtt{\,{\Huge{\rightthreetimes}}\,}
\def\rttt{\,{\underline{\rightthreetimes}}\,}
\def\xra{\xrightarrow}
\def\hra{<}
\def\Ker{{\text{Ker}\,}}
\def\Imm{{\text{Im}\,}}
\def\ZZ{\Bbb Z}
\def\Z{\Bbb Z}
\def\nto{\not\to}
\def\ZpZ{{\Bbb Z}/\!p{\Bbb Z}}
\def\Ab{\textit{Ab}}\def\AbKer{{\textit{AbKer}}}
%\def\lr{{()}} \def\rl{{)(}}
%\def\rr{{))}}
\def\lr{{\rtt lr}} \def\rl{{\rtt rl}}
\def\rr{{\rtt rr}}
\begin{document}
\title[%The Feit-Thompson theorem in the category of finite groups
Basic finite group theory via the lifting property
]{
%Expressing the statement of the Feit-Thompson theorem
%with diagrams in the category of finite groups
%Formulating
Formulating basic notions of \\ finite group theory\\ via the lifting property}
\author[masha gavrilovich]{masha gavrilovich
\thanks{\tiny Institute for Regional Economic Studies, Russian Academy of Sciences (IRES RAS).
National Research University Higher School of Economics, Saint-Petersburg.
%The Higher School of Economics, ???
{\tt \tiny mi\!\!\!ishap\!\!\!p@sd\!\!\!df.org \url{http://mishap.sdf.org}}.\\
This paper commemorates the centennial of the birth of N.A.~Shanin,
the teacher of S.Yu.Maslov and G.E.Mints, who was my teacher. I hope the motivation behind this paper is in spirit of the Shanin's group ТРЭПЛО (теоритическая разработка эвристического поиска логических обоснований, theoretical development of heuristic search for logical evidence/arguments).}
}%%%\thanks
%%{
%% http://mishap.sdf.org/mints-lifting-property-as-negation\newline
%%%National Research University
%%%Higher School of Economics,
%%%Soyza Pechatnikov str., 16, St.Petersburg, Russia. \newline
%%%St. Petersburg Institute for Economics and Mathematics of
%%%the Russian Academy of Sciences.
%%%Serpuhovskaya str., 38, St. Petersburg, Russia.
%%}
%%}
%%%%%
%%%%%\title[Expressing the statement of Feit-Thomson theorem
%%%%%with diagrams in the category of finite groups
%%%%%\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\textbf{\tiny To} {\textsf{\bf Grigori Mints Z"L}} \tiny In memoriam}
%%%%%]{Expressing the statement of Feit-Thomson theorem
%%%%%with diagrams in the category of finite groups
%%%%%\\{\tiny.}
%%%%%\\
%%%%%{{{\tiny To} {\textsf{Grigori Mints Z"L}} \tiny In memoriam}}
%%%%%}
%%%%%\author[Misha Gavrilovich]{Misha Gavrilovich\ \ \ \ \ \
%%%%%{{{\tiny } {\text{July 2016}}}}}
%%%%%\date{2013}
%%%%%
%%%\extraline{\copyright\ 2014 2015 Misha Gavrilovich {\tt mi\!\!\!ishap\!\!\!p@sd\!\!\!df.org http://mishap.sdf.org/mints-lifting-property-as-negation}}
%%%\classno{97A80, 97B70}
%%%\journal{ mishap.sdf.org/mints \hfill
%%%\hfill ISSN 2053-1451}
%%%\volume{5}
%%%\parskip 5pt
%%%\setcounter{page}{23}
%%\date{}
%%%\maketitle
%%
%%\setlength{\epigraphwidth}{0.5\textwidth}
%%\epigraph{\normalsize There's no point in being grown up if you can't be childish sometimes
%%}{ {\emph{ %fixme: a better epigraph is needed.
%%}}
%%}
%%
%%
\begin{abstract}
%\normalsize
%We reformulate the statement of the Feit-Thompson theorem
%in terms of diagrams in the category of finite groups,
%namely
We reformulate several basic notions of notions in finite group theory in terms
of iterations of the lifting property (orthogonality) with respect to
particular morphisms. Our examples include the notions %of a finite group
being nilpotent,
solvable, perfect, torsion-free; p-groups and prime-to-p-groups; Fitting
subgroup, perfect core, p-core, and prime-to-p core.
We also reformulate as in similar terms the conjecture that a localisation of a
(transfinitely) nilpotent group is (transfinitely) nilpotent.
\end{abstract}
\maketitle
%%\Large
%%
%%
\section{Introduction.}
%%We illustrate the generative power of the Quillen lifting property (orthogonality)
%%as a means of
%%defining natural elementary mathematical concepts
%by giving a number of
%examples in various categories,
%in particular
%%by showing that \bi\item several %many
We observe that several
standard
elementary notions of finite group theory %abstract topology
can be defined by
iteratively applying the same diagram chasing ``trick'',
namely the lifting property (orthogonality of morphisms),
to simple classes of homomorphisms of finite groups.
%in the category of finite groups,
%the ``trick''
%is known as the Quillen lifting property and was introduced
% by Quillen [Qui] to axiomatise algebraic topology
%in terms of categories.
%
%applying the Quillen lifting
%property to simple classes of morphisms of finite groups. %topological spaces.
%\ei
%Examples in topology include the notions of: compact, discrete, connected, and
%totally disconnected spaces, dense image, induced topology, and separation
%axioms. Examples in algebra include:
The notions include a finite group being nilpotent, solvable, perfect,
torsion-free; $p$-groups, and prime-to-$p$ groups; $p$-core, the Fitting subgroup, cf.\S\ref{sec:neg}-\ref{sec:reforms}.
%; injective and projective
%modules; injective, surjective, and split homomorphisms.
%This is stated in Theorem~\ref{theo:1}, Corollary~\ref{coro:1}~and~\ref{coro:2}.
In \S\ref{sec:locals} we reformulate as a labelled commutative diagram the conjecture %of Farjoun
that a localisation
of a transfinitely nilpotent group is transfinitely nilpotent; this suggests a variety of related questions
and is inspired by the conjecture of Farjoun that a localisation of a nilpotent group is nilpotent.
The goal of this paper to present a collection of examples which show
the lifting property is all that's needed to be able to
define a number of notions from simplest (counter)examples of interest.
Curiously, our observations lead to a concise and uniform notation (Theorem~\ref{theo:1}, Corollary~\ref{coro:1}~and~\ref{coro:2}), e.g.
$$ (\ZpZ\lra 0)^{rr},\ \ (\AbKer)^{lr},\ \text{ and }\ (0\lra *)^{lr} $$
denote the classes of homomorphisms (of finite groups)
whose kernel is a $p$-group, resp.~soluble, subgroup, and those corresponding to subnormal subgroups.
One might hope that a notation so concise and uniform might be of use in computer algebra and automated theorem provers.
Deciphering this notation can be used as an elementary exercise in a first course of group theory or category theory on basic definitions and diagram chasing.
%e.g.~subnormal subgroups are denoted by $(0\lra *)^{lr}$, %denotes the class of subnormal groups,
%and $p$-groups are denoted by $(\ZpZ\lra 0)^{rr}$ or $(0\lra \ZpZ)^{lr}$,
%and $(\AbKer)^{lr}$ denotes the class of homomorphisms with soluble kernel where $(\AbKer)$
%denotes the class of homomorphisms with Abelian kernel.
%We include some speculations on the wider significance of this, particularly in
%formalisation of mathematics. A draft. As a text file (more convenient for some
%readers).
%Is this merely a coincidence? .... calculate more examples of iterated Quillen negations...
Such reformulations lead one to the following questions:
\bi\item
Can one extend this notation to capture more of finite group theory?
\item Is this a hint towards
category theoretic point of view on finite group theory?
%Is this merely a coincidence, or is it a hint that a significant part of finite group theory
%admits a concise reformulation in terms of elementary category theory, more particularly
%diagram chasing and Quillen lifting property (orthogonality), which
%can be of use in computer algebra and automated theorem proving?
\ei
If one believes the evidence
provided by our examples %mine
is strong enough to demand an explanation, then
one should perhaps start by trying to find more examples defined in this way, and
by calculating the classes of homomorphisms obtained by
iteratively applying the Quillen lifting
property to simple classes of morphisms of finite groups.
%Note that our reformulations are very concise and uniform;
%therefore one may consider whether they can be of use in in computer algebra and automated theorem proving.
%We hope our reformulations can be used as elementary exercises in a first course of group theory or category theory.
%has to seek an appropriate mathematical context where
%iterated Qullien negations %such powers of $\pi$
%occur.
%
%
%
%
%\begin{enonce}{Question} Is there a category theoretic description %of the sort
%which leads to a concise reformulation of finite group theory
%and can be of use in proof verification, computer algebra, and automated theorem proving?
%\end{enonce}
%
%If one belives the evidence
%provided by our examples %mine
%is strong enough to demand an explanation, then
%one has to seek an appropriate mathematical context where
%iterated Qullien negations %such powers of $\pi$
%occur.
%
%
%If one belives the evidence
%provided by our examples %mine
%is strong enough to demand an explanation, then
%one has to seek an appropriate
%mathematical context where
%iterated Qullien negations %such powers of $\pi$
%occur. ????
%
%
%
%
%
%
%
%We reformulate several %a few|several? basic?
% notions in finite group theory
%in terms of diagrams in the category of finite groups,
%namely being solvable, nilpotent, $p$-group and prime-to-$p$ group,
%abelian, perfect, subnormal subgroup, injective and surjective homomorphism
%(see Fig.~1 and Fig.~2).
%These notions are enough to formulate the Feit-Thompson theorem.
%
%
%These properties are obtained by iterating the same diagram chasing ``trick''
%%, the Quillen lifting property,
%in the category of groups,
%often starting from a single morphism-counterexample; the ``trick''
%is known as the Quillen lifting property and was introduced
% by Quillen [Qui] to axiomatise algebraic topology
%in terms of categories.
%
\subsubsection*{Motivation} Our motivation was to formulate %the Feit-Thompson theorem
part of finite group theory in a form amenable
to automated theorem proving while remaining human readable;
[GP] tried to do the same thing for the basics of general topology.
%; given a class (property) $P$ of morphisms in a category,
%a diagram chasing ``trick'', the Quillen lifting property (see Figure 1)
%this trick defines two classes $P^{\rtt l}$ and $P^{\rtt r}$ which intersect $P$ only by isomorphisms.
%These diagrams are surprisingly uniform; the same diagram is used repeatedly, chasing properties are repeated applications of the same diagram chasing ``trick'',
%the Quillen lifting property, to the simplest morphism-counterexample
%or previously obtained lifting properties. This suggest that perhaps these reformulations
%may be used by an automated theorem prover.
Little attempt has been made to go beyond these examples.
Hence open questions remain: are there other interesting examples of lifting properties in the category of (finite) groups?
Can a complete group-theoretic argument be reformulated in terms of diagram chasing, say
the classification of CA-groups or $pq$-groups, or
elementary properties of subgroup series; can category theory notation
be used to make expositions easier to read?
Can these reformulations be used in automatic theorem proving?
Is there a decidable fragment of (finite) group theory
based on the Quillen lifting property and, more generally,
diagram chasing, cf.~[GLS]?
Can the Sylow theorems (only existence and uniqueness of Sylow subgroups)
be proven using this characterization of p-groups?
Could the components of a finite group, and their properties (commute
pairwise, commute with normal p-subgroups) be characterized and proven to exist with
these methods?
\section{Definitions and examples of reformulations}
\subsection{Key definition: the Quillen lifting property (negation/orthogonal)}
The Quillen lifting property, also known as orthogonality of morphisms, is a property of a pair of morphisms in a category.
It appears in a prominent way
in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen,
and %is used in homotopy
%theory within algebraic topology
is used to define properties of morphisms starting from an explicitly given class of morphisms.
%Using the Quillen lifting property is perhaps the simplest way to define a class of morphisms {\em without} a property
%in a manner useful in a category theoretic diagram chasing computation.
%It appears in a prominent way
% in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen.
%
%Taking Quillen negation/orthogonal is perhaps the simplest way to define a class of morphisms {\em without} a property
%in a matter useful in a category theoretic diagram chasing computation:
\begin{defi} We say that two morphisms $A \xra f B$ and $X \xra g Y$ in a category $C$ are {\em orthogonal} and write $f\rtt g$
iff
for each $i:A\lra X$ and $j:B\lra Y$
making the square commutative, i.e.~$f\circ j=i\circ g$ there is a diagonal arrow $\tilde j:B\lra X$ making the total diagram
$A\xra f B\xra {\tilde j} X\xra g Y, A\xra i X, B\xra j Y$ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$ (see Figure 1a).
We may also say that {\em $f$ lifts wrt $g$}, {\em $f$ left-lifts wrt $g$}, or {\em $g$ right-lifts wrt $f$}, or that $f$ {\em antagonizes} $g$.
%Let $P$ be a class (property) of mophisms in $C$. %Its {\em left and right Quillen negation} is
%Denote
% $$ P^{\rtt l}:=\{\, p^l\ :\ p^l\,\rtt\,p\,\ \text{for each }p\in P\,\}$$
%$$P^{\rtt r}:=\{\, p^{\rtt r}\ :\ p\, \rtt p^{\rtt r}\,\ \text{for each }p\in P\,\}$$
%
%We call $ P^{\rtt l}$ {\em left Quillen negation} or {\em left Quillen orthogonal} of $P$, and
%we call $ P^{\rtt l}$ {\em right Quillen negation} or {\em right Quillen orthogonal} of $P$.
%\end{defi}
By analogy with orthogonal complement of a non-symmetric bilinear form, % as remarked by Maxim
define {\em left/right Quillen negation} or {\em left/right Quillen orthogonal} of a class $P$
of morphisms:
$$ P^{\rtt l}:=\{\, f\ :\ f\,\rtt\,g\,\ \text{for each }g\in P\,\}$$
$$P^{\rtt r}:=\{\, g\ :\ f\, \rtt g\,\ \text{for each }g\in P\,\}$$
\end{defi}
\vskip 3pt
We have
$$ P^{\rtt l}=P^{\rtt lrl},\ P^{\rtt r}=P^{\rtt rlr},\ P\subset P^{\rtt lr},\ P\subset P^{\rtt rl}$$
$$P \subset Q \text{ implies } Q^{\rtt l} \subset P^{\rtt l},\ Q^{\rtt r} \subset P^{\rtt r},\ P^{\rtt lr} \subset Q^{\rtt lr},\ P^{\rtt rl} \subset Q^{\rtt rl} $$
$$ P\cap P^{\rtt l} \subset (Isom),\ \ P\cap P^{\rtt r}\subset (Isom)$$
Under certain assumptions on the category and property $P$
Quillen small object argument shows that each morphism $G\lra H$ decomposes both as
$$G \xra{(P)^{\rtt lr}}\cdot \xra{(P)^{\rtt r}} H \text{ and }
G \xra{(P)^{\rtt r}}\cdot \xra{(P)^{\rtt lr}} H .$$
Using the Quillen lifting property is perhaps the simplest way to define a class of morphisms {\em without} a given property
in a manner useful in a category theoretic diagram chasing computation.
\def\rrt#1#2#3#4#5#6{\xymatrix{ {#1} \ar[r]^{} \ar@{->}[d]_{#2} & {#4} \ar[d]^{#5} \\ {#3} \ar[r] \ar@{-->}[ur]^{}& {#6} }}
\begin{figure}
\begin{center}
\large
$(a)\ \xymatrix{ A \ar[r]^{i} \ar@{->}[d]_f & X \ar[d]^g \\ B \ar[r]_-{j} \ar@{-->}[ur]^{{\tilde j}}& Y }$% \
$\ \ \ \ \ \ (b)\ \xymatrix{ A \ar[r] \ar@{->}[d]_{(P)} & X \ar[d]^{\therefore (Q)} \\ B \ar[r] \ar@{-->}[ur]& Y }$% \
$\ \ \ \ \ \ (c)\ \xymatrix{ A \ar[r] \ar@{->}[d]_{\therefore(P)} & X \ar[d]^{ (Q)} \\ B \ar[r] \ar@{-->}[ur]& Y }$% \
%$\rrt ABXY$\ \ \ \
%$(b)\ \rrt {\{\}} {} {\{\bullet\}} X {\therefore(surj)} Y $%\
%$(c)\ \rrt {\{\bullet,\bullet\}} {} {\{\bullet\}} X {\therefore(inj)} Y $%\ \
%$(d)\ \rrt X {\therefore(inj)} {Y} {\{x,y\}} {} {\{x=y\}}$\
\end{center}
\caption{\label{fig1}\small
%Lifting properties. Dots $\therefore$ indicate free variables and what property of these variables is being defined;
%, i.e.~a property of what is being defined and how is it to be labelled
%in a diagram chasing calculation, ``$\therefore(surj)$" reads as:
%given a (valid) diagram, add label $(surj)$ to the corresponding arrow.\newline
(a)
The definition of a lifting property $f\rtt g$: for each $i:A\lra X$ and $j:B\lra Y$
making the square commutative, i.e.~$f\circ j=i\circ g$, there is a diagonal arrow $\tilde j:B\lra X$ making the total diagram
$A\xra f B\xra {\tilde j} X\xra g Y, A\xra i X, B\xra j Y$ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$.
We say that $f$ lifts wrt $g$, $f$ left-lifts wrt $g$, or $g$ right-lifts wrt $f$.
\ (b) Right Quillen negation.
The diagram defines a property $Q$ of morphisms in terms of a property $P$; a morphism has property (label) $Q$ iff it right-lifts
wrt any morphism with property $P$, i.e.
$Q=\{\, q :\ p\, \rtt q\,\ \text{for each }p\in P\,\}$
\ (c) Left Quillen negation.
The diagram defines a property $P$ of morphisms in terms of a property $Q$; a morphism has property (label) $P$ iff it left-lifts
wrt any morphism with property $Q$, i.e.
$ P=\{\, p\ :\ p\,\rtt\,q\,\ \text{for each }q\in Q\,\}$
% (b) $X\lra Y$ is surjective\newline
% (c) $X\lra Y$ is injective; $X\lra Y$ is an epicmorphism if we forget %never use
%that $\{\bullet\}$ denotes a singleton (rather than an arbitrary object
%and thus $\{\bullet,\bullet\}\lra\{\bullet\}$ denotes an arbitrary morphism $Z\sqcup Z\xra{(id,id)} Z$)\newline
% (d) $X\lra Y$ is injective, in the category of Sets; $\pi_0(X)\lra\pi_0(Y)$ is injective,
% when the diagram is interpreted in the category
%of topological spaces.
}
\end{figure}
\subsection{\label{sec:neg}A list of iterated Quillen negations of simple classes of morphisms}
Let $(0\lra *)$, resp.~$(0\lra Ab)$, denote the class of morphisms from the trivial groups to an arbitrary group, resp. Abelian group.
Let $(*\lra 0)$, resp.~$(Ab\lra 0)$, denote the class of morphisms to the trivial groups from an arbitrary group, resp. Abelian group.
Let $(AbKer)$ denote the class of homomorphisms with an Abelian kernel.
\begin{theo}\label{theo:1}
In the category of Finite Groups,
\begin{enumerate}
\item $(\AbKer)^\lr$ is the class of homomorphisms whose kernel is solvable
\item $(0\lra *)^\lr$ is the class of subnormal subgroups
\item $(0\lra \Ab)^\lr=(0\lra \Ab)^\lr=\{[G,G]\lra G: G\text{ is arbitrary}\}^\lr$ is the class of subgroups $H)^{rl}$
\item $(\AbKer)^{\rtt l}$ is the class of homomorphisms whose kernel is perfect
\end{enumerate}
\end{theo}
\begin{proof} The proof is a matter of deciphering notation.
Proof of item 1. First note that $P\lra 0\in (\AbKer)^{\rtt l}$ for a perfect group, and $P\lra
0 \rtt H\lra G$ implies $\Ker(H\lra G)$ is soluble. This means that $(\AbKer)^{\rtt lr}$
is contained in the class of maps whose kernel is soluble. On the other hand,
any such map is a composition of maps $H/[S_n,S_n]\lra H/S_n$, ..., $H/[S_0,S_0]\lra H/S_0$, and $\Imm H\lra G$ where $S_{n+1}=[S_n,S_n],S_0=\Ker(H\lra G)$
is the descending derived series.
Now let us prove item 2.
By definition, $A\lra B$ is in $(0\lra *)^{\rtt l}$ iff $A\lra B
\rtt 0\lra G $ for any group $G$. Take $G=B/A^B$ to be the quotient of $B$ by
the normal closure of $A$, and $B\lra G$ to be the quotient map. This shows
that if $G=B/A^B$ is non-trivial, then the lifting property fails.
On the other hand, it is easy to check the lifting property holds that in a commutative square,
the map to $G$ factors via $B/A^B$, hence the lifting property holds if $B/A^B$ is trivial.
Let $C \vartriangleleft D$ be a normal subgroup. The lifting property $A\lra B
\rtt 0\lra D/C $ implies $A\lra B
\rtt C\lra D $. Orthogonals are necessarily closed under composition, hence this implies that
if $C$ is a subnormal subgroup of $D$, i.e.~there exists a series
if $C \vartriangleleft D_n \vartriangleleft D_{n-1} \vartriangleleft ... \vartriangleleft D_1 \vartriangleleft D$,
then the lifting property holds and $C\lra D$ is in $(0\lra *)^{lr}$.
Now assume that $C$ is not subnormal in $D$ and let $C0 \}^{\rtt r}$
\item a subgroup \ensuremath{H0 \}^{\rtt r}$
\item $H$ is a verbal subgroup of $G$ generated by substitutions in words $w_1,..,w_i$ in the free group $\Bbb F_n$ iff $H$ fits into an exact sequence
$$H \lra G \xra {(\Bbb F_n \lra\, \Bbb F_n/)^{\rtt rl}} \cdot \xra{(\Bbb F_n \lra\, \Bbb F_n/)^{\rtt r}} 0,$$ or, equivalently, is the kernel of the corresponding homomorphism
\item a free Burnside group of power $n$
is a group $B$ which fits into a decomposition of form
$$ 0\xra{(0\longrightarrow \ZZ)^{rl}} \cdot \xra {(\Bbb Z \lra\, \ZZ/n\ZZ )^{\rtt rl}} B \xra{(\ZZ \lra \ZZ/n\ZZ)^{\rtt r}} 0,$$
\item a group $S$ is transfinitely soluble,
i.e.~there exists an ordinal $\alpha$ such that $G^\alpha=0$, where $G^{\beta+1}=[G,G^\beta]$, and $G^\beta=\cap_{\gamma<\beta} G^\gamma$ whenever $\beta\neq \gamma+1$ for $\gamma<\beta$,
iff %either of the following equivalent conditions holds:
\bi\item $S\lra 0$ is in $(\AbKer)^\lr$
\ei
\item a group $G$ is transfinitely nilpotent, i.e.~there exists an ordinal $\alpha$ such that $G^\alpha=0$, where $G^{\beta+1}=[G,G^\beta]$, and $G^\beta=\cap_{\gamma<\beta} G^\gamma$ whenever $\beta\neq \gamma+1$ for $\gamma<\beta$,
iff
\begin{itemize}\item the diagonal map $H\lra H\times H$, $x\mapsto (x,x)$, is in $(0\lra *)^\lr$
\end{itemize}
\end{enumerate}
\end{coro}
\begin{coro}%[Feit-Thompson]
The statement that a group of odd order is necessarily soluble is represented by either of the following inclusions
$$ (\ZZ/2\ZZ\longrightarrow 0)^{\rtt l} \subset (\AbKer)^\lr
$$
$$ (2\ZZ\lra \ZZ)^{\rtt r} \cap (0\lra *)^\lr \subset (0\lra \Bbb Q/\ZZ)^\lr
$$
calculated in the category of Finite Groups.
\end{coro}
%
%
%$ 0 \xra{(0\lra \ZZ/2\ZZ)^{\rtt r}} O_{2'}(G) \lra \xra o$
%
%$\lra \xra{(0\lra *)^{\rtt l}} G\cdot \xra{(0\lra *)^\lr} G$
%
%decompsotion
\subsection{$p$-, $p'$-, and $p,p'$-core as an example of a weak factorisation system}
Axiom M2 of a Quillen model category requires that each morphism $A\lra B$ decomposes as
$$ A\xra{(c)} \cdot \xra{(f)} B$$
where $(c)$ and $(f)$ are orthogonal to each other.
These decomposition give rise to weak factorisation systems whose existence is proven by the Quillen small object argument.
There are somewhat similar decompositions in group theory.
That ``each group admits a surjection from a free group'' can be denoted as follows; each morphism $0\lra G$ admits a decomposition
$$0\xra{(0\longrightarrow \ZZ)^{rl}} \cdot \xra{(0\longrightarrow \ZZ)^{r}} G $$
in this notation, we think of the Quillen orthogonals as {\em labels} put on arrows, hence
the notation means that the homomorphisms belong to the corresponding Quillen orthogonals.
In a finite group, the descending derived series stabilises at a perfect subgroup $P=[P,P]$ (its perfect core) which is characteristic,
corresponds to the unique decomposition of form
$$ H \xra{(\AbKer)^{\rtt l}} \cdot \xra{(\AbKer)^{\rtt lr}} G
$$
of a morphism into a map with a perfect kernel $P$, and a map with a soluble kernel.
Note these decompositions are analogous to decompositions appearing in weak factorisation systems proved by the Quillen small object argument.
\begin{coro}[$p$-core, $p'$-core, $p,p'$-core]\label{coro:2} In the category of Finite Groups,
\begin{itemize}
\item the $p$-core $O_{p}(G)$ of $G$, i.e.~the largest normal $p$-subgroup of $G$,
is the group appearing in the unique decomposition of form
$$ G \xra {(\ZpZ\lra 0)^\rr} G/O_p(G) \xra{(\ZpZ\lra 0)^{\rtt rrl}} 0$$
% 0 \xra{(0\lra \ZZ/2\ZZ)^{\rtt r}} O_{2'}(G) \lra \xra{(0\lra *)^\lr} G $$
\item the $p'$-core $O_{p'}(G)$ of $G$, i.e.~the largest normal $p'$-subgroup of $G$,
is the group appearing in the unique decomposition of form
$$ G \xra {(\ZpZ\lra 0)^{\rtt r}} G/O_{p'}(G) \xra{(\ZpZ\lra 0)^{\rtt rl}} 0$$
\item the $p,p'$-core $O_{p,p'}(G)=O_p(G/O_{p'}(G))$ of $G$ %, i.e.~the largest normal $p'$-subgroup of $G$,
is the group appearing in the unique decomposition of form
$$ G \xra {(\ZpZ\lra 0)^{\rtt r}} G/O_{p'}(G) \xra {(\ZpZ\lra 0)^{\rtt rr}} G/O_p(G/O_{p'}(G)) \xra{(\ZpZ\lra 0)^{\rtt rl}} 0$$
\end{itemize}
\end{coro}
We end with a couple of test questions suggested by Bob Oliver (private communication);
see \S\ref{sec:sylow} for some suggestions.
\begin{que}[Bob Oliver]
Can the Sylow theorems (only existence and uniqueness of Sylow subgroups)
be proven using the characterization of p-groups by Corollary~\ref{coro:1}(4) ?
Could the components of a finite group, and their properties (commute
pairwise, commute with normal p-subgroups) be characterized and proven to exist with
help of our reformulations?
\end{que}
\subsection{$f$-local groups, localisations and nilpotent groups.\label{sec:locals}}
\footnote{We thank S.O.Ivanov for pointing out the notion of $f$-local groups and the conjecture of Farjoun
that a localisation of a nilpotent group is nilpotent [AIP].}
\def\urtt{\, !\!\rightthreetimes\,}
\def\uurtt{\,!\!\!\!\rightthreetimes\,}
\def\Groups{{\text{Groups}}}
\def\Id{{\text{Id}}}
%This is a preliminary unproofread section whose purpose is to collect references and comments received in response to the note.
%
%Comments below follow a discussion with S.O.Ivanov.
Let $ f \uurtt g$ denote the {\em unique} lifting property.
For a morphism $f$ of groups, a group $A$ is called $f$-local iff $ f \uurtt A\lra 0$. Under some assumptions,
each morphisms $H\xra g G$ of groups decomposes as
$$ H \xra{(f)^{\urtt rl}} \cdot \xra{(f)^{\urtt r}} G$$
A diagram chasing argument shows that whenever such a decomposition always exists,
there is a functor $L=L_f: \Groups\lra \Groups $ defined by
$$H \xra{(f)^{\urtt rl}} L(H) \xra{(f)^{\urtt r}} 0,$$
a natural transformation $\eta: \Id \lra L: \Groups\lra\Groups$ which induces
isomorphisms $\eta_G:L\, G \xra{(iso)} LL\, G$, $\eta_{LG}=L(G\xra{\eta_X}LG): L\,G\lra LL\,G$.
A functor with these data is called an {idempotent monad} or a {\em localisation}, and by [CSS] Vopenka principle implies
that any localisation is of this form. See [AIP] for details and references.
Our notation allows to express a property closely related to the conjecture
of Farjoun
that
the localisation of a nilpotent group is nilpotent, as follows;
see [AIP] and references therein for a discussion of this conjectures.
Note the diagram has a symmetry: it mentions the diagonal map $H\lra H\times H$.
\begin{conj}[Farjoun]
The following diagram holds for any property (class) of homomorphisms $L$.
$$
\xymatrix@C+2pc{ {H} \ar[r]^{(L)^{\urtt rl}} \ar@{->}[d]|{(0\lra *)^{\urtt lr}} & {H_L} \ar[d]|{\therefore(0\lra *)^{\urtt lr}} \ar[r]^-{{(L)^{\urtt r}}} & 0 \ar[d] \\
{H\times H} \ar[r]^{(L)^{\urtt rl}} & {H_L\times H_L} \ar[r]^--{{(L)^{\urtt r}}} & 0
}
\ \ \ \ \xymatrix@C+2pc{ {H} \ar[r]^{(L)^{\urtt rl}} \ar@{->}[d]|{(0\lra \Ab)^{\urtt lr}} & {H_L} \ar[d]|-{\therefore(0\lra \Ab)^{\urtt lr}} \ar[r]^-{{(L)^{\urtt r}}} & 0 \ar[d] \\
{H\times H} \ar[r]^{(L)^{\urtt rl}} & {H_L\times H_L} \ar[r]^--{{(L)^{\urtt r}}} & 0
}
$$
In the diagram above, ``$\therefore(label)$" reads as:
given a (valid) diagram whose arrows have properties indicate by their labels, the arrow marked by $\therefore$ has the property indicated by its label.
See~Fig.~1 and Corollary~2.3(2) for explanations and details.
\end{conj}
Our notation suggests the following modifications of the conjecture.
\begin{que} Does it hold for each morphism $H\lra G$ of groups and any
homomorphism $f$:
$$\xymatrix@C+5pc{ {H} \ar[r]^{(f)^{\urtt rl}} \ar@{->}[d]|{(0\lra \Ab)^{\urtt lr}} & {H_f} \ar[d]|-{\therefore(0\lra \Ab)^{\urtt lr}} \ar[r]^-{{(f)^{\urtt r}}} & 0 \ar[d] \\
{G} \ar[r]^{(f)^{\urtt rl}} & {G_f} \ar[r]^--{{(f)^{\urtt r}}} & 0
}
$$
\end{que}
\begin{que} Does it hold for any diagonal morphism $H\lra H\times H$ of groups, any
properties (classes) $L$ and $P$ of homomorphisms:
$\xymatrix@C+3pc{ {H} \ar[r]^{(L)^{\urtt rl}} \ar@{->}[d]|{(P)^{\urtt lr}} & {H_L} \ar[d]|-{\therefore(P)^{\urtt lr}} \ar[r]^-{{(L)^{\urtt r}}} & 0 \ar[d] \\
{H} \ar[r]^{(L)^{\urtt rl}} & {H\times H_L} \ar[r]^--{{(L)^{\urtt r}}} & 0
}
$
$\xymatrix@C+3pc{ {H} \ar[r]^{(L)^{rl}} \ar@{->}[d]|{(P)^{lr}} & {H_L} \ar@{..>}[d]|-{\exists\,(P)^{lr}} \ar[r]^-{{(L)^{r}}} & 0 \ar[d] \\
{H} \ar[r]^{(L)^{rl}} & {H\times H_L} \ar[r]^--{{(L)^{r}}} & 0
}
$
\end{que}
\begin{que} Under what assumptions on morphism $f:H\lra G$, properties $L$ and $P$
of homomorphisms it holds:
$
\xymatrix@C+3pc{ {H} \ar[r]^{(L)^{\urtt rl}} \ar@{->}[d]_{(P)^{\urtt lr}} & {H_L} \ar[d]|{\therefore(P)^{\urtt lr}} \ar[r]^-{{(L)^{\urtt r}}} & 0 \ar[d] \\
{G} \ar[r]^{(L)^{\urtt rl}} & {G_L} \ar[r]^--{{(P)^{\urtt r}}} & 0
}$
$\ \ \ \ \ \
\xymatrix@C+3pc{ {H} \ar[r]^{(L)^{\rtt rl}} \ar@{->}[d]_{(P)^{\rtt lr}} & {H_L} \ar@{..>}[d]|{\exists\,(P)^{\rtt lr}} \ar[r]^-{{(L)^{\rtt r}}} & 0 \ar[d] \\
{G} \ar[r]^{(L)^{\rtt rl}} & {G_L} \ar[r]^--{{(P)^{\rtt r}}} & 0
}$
\end{que}
In an obvious way the notation suggests a large number of similar questions.
The following is only an example, there is little motivation for this particular choice.
We use this example as an opportunity to use shortened notation.
\begin{que} Under what assumptions on %morphism $f:H\lra G$,
properties $\Delta$, $L$ and $P$
of homomorphisms it holds:
$$
\xymatrix@C+3pc{ {\cdot} \ar[r]^{(L)^{\urtt r..rl}} \ar@{->}[d]_{(P)^{\urtt l..lr}}^{(\Delta)} & {\cdot} \ar[d]|{\therefore(P)^{\urtt l..lr}} \ar[r]^-{{(L)^{\urtt r..r}}} & \cdot \ar[d]|{(\Delta)} \\
{\cdot} \ar[r]|{(L)^{\urtt r..rl}} & {\cdot} \ar[r]|--{{(P)^{\urtt r..r}}} & \cdot
}
\ \ \ \ \ \
\xymatrix@C+3pc{ {\cdot} \ar[r]^{(L)^{\rtt r..rl}} \ar@{->}[d]_{(P)^{\rtt l..lr}}^{(\Delta)} & {\cdot} \ar@{..>}[d]|{\exists\,(P)^{\rtt l..lr}} \ar[r]^-{{(L)^{\rtt r..r}}} & \cdot \ar[d]|{(\Delta)} \\
{\cdot} \ar[r]|{(L)^{\rtt r..rl}} & {\cdot} \ar[r]|--{{(P)^{\rtt r..r}}} & \cdot
}$$
In particular, when does it hold for $\Delta=\{H\lra H\times H\,:\,H\text{ a group}\}$ the class of diagonal embeddings?
\end{que}
\subsection{Reformulations with less notation}
In this subsection in a verbose manner we decipher the notation of Quillen negation of the examples below.
Fig.~3 represents considerations below as diagrams.
There is no non-trivial homomorphism from a group $F$ to $G$, write $F\nto G$, iff
$$0\lra F\,\rtt 0\lra G\text{ or equivalently }F\lra 0 \rtt G\lra 0.$$
A group $A$ is {\em Abelian} iff
\[ \left \,\lra\,\left \rtt\,\, A\lra 0
\]
where $\left \,\lra\,\left$ is the abelianisation morphism sending the free group into the Abelian free group on two generators;
a group $G$ is {\em perfect}, $G=[G,G]$, iff $G\nto A$ for any Abelian group $A$, i.e.
\[ \left \,\lra\,\left \rtt\,\, A\lra 0\ \implies\ G\lra 0 \rtt A\lra 0\]
equivalently, for an arbitrary homomorphism $g$,
\[ \left \,\lra\,\left \rtt\,\,g \ \ \implies\ G\lra 0 \rtt\, \,g\,\]
Yet another reformulation is that, for each group $S$,
$$
0\lra G \,\rtt\, [S,S]\lra S .$$
In the category of finite or algebraic groups,
a group $H$ is {\em soluble} iff $G\nto H$ for each perfect group $G$,
i.e.
$$0\lra G\,\rtt 0\lra H\text{ or equivalently }G\lra 0 \rtt H\lra 0.$$
Alternatively, a group $H$ is {\em soluble} iff for every homomorphism $f$ it holds
$$ f \,\rtt\, [G,G]\lra G \text{ for each group }G\ \implies\ f \,\rtt 0\lra\, H
.$$
A prime number $p$ does not divide the number elements of a finite group $G$
iff $G$ has no element of order $p$, i.e. no element $x\in G$ such that $x^p=1_G$ yet $x^1\neq 1_G,...,x^{p-1}\neq 1_G$,
equivalently $\ZpZ\nto G$, i.e.
$$0\lra \ZpZ\,\rtt 0\lra G\text{ or equivalently }\ZpZ\lra 0 \rtt G\lra 0.$$
A finite group $G$ is a $p$-group, i.e. the number of its elements is a power of a prime number $p$, iff in the category of finite groups
$$0\lra \ZpZ \rtt 0\lra H \implies 0\lra H\rtt 0\lra G.$$
A group $H$ is the normal closure of the image of $N$,
i.e.~no proper normal subgroup of $H$ contains the image of $N$, iff for an arbitrary group $G$
$$N\lra H \,\rtt 0 \lra G.$$
A group $D$ is a subnormal subgroup of a finite group $G$ iff
$$ N\lra H \,\rtt 0 \lra B \text{ for each group }B\ \implies N\lra H\,\rtt\,D \lra G
$$
i.e. $D\lra G$ right-lifts wrt any map $N\lra H$ such that $H$ is the normal closure of the image of $N$; the lifting property implies that $D\lra G$ is injective. Recall that $D$ is a subnormal subgroup of a finite group $G$ iff there is a finite series of subgroups
%$$D=G_0< G_1 < \ldots <\normal G_n =G$$
$$D=G_0 \vartriangleleft G_1 \vartriangleleft \ldots \vartriangleleft G_n =G$$
such that $G_i$ is normal in $G_{i+1}$, $i=0,\ldots,n-1$.
This is probably the only claim which requires a proof. First notice that if $D$ is normal in $G$ then
the lifting property holds.
Given a square corresponding to $N\lra H \,\rtt\, D\lra G$, the preimage of $D$ in $H$
is a normal subgroup of $H$ containing the image of $N$, hence the preimage of $D$ contains $H$ and the lifting property holds.
The lifting property is closed under composition, hence it holds for subnormal subgroups as well.
Now assume $D$ is not subnormal in $G$. As $G$ is finite, there is a minimal subnormal subgroup $D'>D$ of $G$.
By construction %$D'$ is the normal closure of $D$ in $D'$
no proper normal subgroup of $D'$ contains $D$
but the lifting property $D\lra D' \,\rtt\, D\lra G$ fails.
Finally, a finite group $G$ is nilpotent iff the diagonal group $G$ is subnormal in $G\times G$ [Nilp], i.e.~iff
the diagonal map $G\xra \Delta G\times G$, $g\mapsto (g,g)$
right-lifts wrt any $N\lra H$ such that $H$ is the normal closure of the image of $N$,
$$ N\lra H \,\rtt 0 \lra B \text{ for each group }B\ \implies N\lra H\,\rtt\, G\xra \Delta G\times G
.$$
\section{Speculations. Extending the notation: Sylow subgroups and normalisers.}
In this section we make some speculation and remarks on ways to extend our notation
to capture the notions of {\em Sylow subgroup} and {\em normaliser} of a subgroup.
\subsection{\label{sec:sylow}Sylow subgroups and diagrams commuting up to conjugations.}
It is useful to consider diagrams which commute {\em up to conjugation}.
Inner automorphisms have the following properties which are useful in a diagram chasing computation,
and which in fact characterise inner automorphisms among all automorphisms [Inn, Sch]:
\bi\item
%We remark that the notion of
%an inner automorphism can be reformulated in a diagram chasing manner.
An inner automorphism of a group $G$ extends along any group homomorphism $\iota: G\lra H$,
i.e.~for any $g\in G$, the inner automorphism $G\lra G, x\mapsto gxg^{-1}$ extends
to an inner automorphism $H\lra H, x\mapsto \iota(g)x\iota(g^{-1})$
\item An inner automorphism of a group $G$ lifts along any surjective group homomorphism $\iota: G\lra H$,
i.e.~for any $g\in G$, the inner automorphism $G\lra G, x\mapsto gxg^{-1}$ extends
to an inner automorphism $H\lra H,\ x\mapsto \iota^{-1}(g)x\iota^{-1}(g^{-1})$
\ei
\begin{figure}
\begin{center}
\large
$(a)\ \xymatrix@C+3pc{ A \ar[r]^{i} \ar@{->}[d]_f & X \ar[d]_{\exists y\in G}^{y gy^{-1}} \\ B \ar[r]|-{j} \ar@{-->}[ur]|{{\tilde j}}& Y }$% \
$(b)\ \xymatrix@C+3pc{ P_1 \ar[r] \ar@{->}[d]|{(0\lra\ZpZ)^{\rttt lr}} & S_p \ar[d]_{\exists y\in G}^{\,\,y gy^{-1}}
\\ P_2 \ar[r] \ar@{-->}[ur] & G }$ \
%$(a)\ \xymatrix{ A \ar[r]^{i} \ar@{->}[d]_f & X\ar@{-->}[r]^\sigma & X^\sigma \ar[d]|--{\sigma g\sigma^{-1}} \\ B \ar[r]|-{j} \ar@{-->}[urr]|{{\tilde j}}& Y \ar@{-->}[r]^\sigma & Y }$% \
%$(b)\ \xymatrix{ P_1 \ar[r] \ar@{->}[d]|{(0\lra\ZpZ)^{\rttt lr}} & S_p\ar@{-->}[r]^\sigma & S_p^\sigma \ar[d]|--{\sigma g\sigma^{-1}}
%\\ P_2 \ar[r] \ar@{-->}[urr] & G \ar[r]^\sigma & G }$ \
$(c)\ \xymatrix@C+2pc{ A \ar[r]|-{(u)} \ar@{->}[d]_f & X \ar[d]|-{g} \\ B \ar[r]|-{(d)} \ar@{-->}[ur]|{{\tilde j}} & Y} $
$(d)\ \xymatrix@C+2pc{ A \ar[r]^{(surj)} \ar@{->}[d]_{(0\lra *)^{\rtt lr}} & N \ar[d] \\ B \ar[r] \ar@{-->}[ur]|{{\tilde j}} & G} $
%$(c)\ \xymatrix{ A \ar[r]-{(surj)} \ar@{->}[d]-{(0\lra *)^{\rtt lr}} & N \ar[d] \\ B \ar[r] \ar@{-->}[ur]|{{\tilde j}}& G }$% \
%$\rrt ABXY$\ \ \ \
%$(b)\ \rrt {\{\}} {} {\{\bullet\}} X {\therefore(surj)} Y $%\
%$(c)\ \rrt {\{\bullet,\bullet\}} {} {\{\bullet\}} X {\therefore(inj)} Y $%\ \
%$(d)\ \rrt X {\therefore(inj)} {Y} {\{x,y\}} {} {\{x=y\}}$\
\end{center}
\caption{\label{fig1}\small
%Lifting properties. Dots $\therefore$ indicate free variables and what property of these variables is being defined;
%, i.e.~a property of what is being defined and how is it to be labelled
%in a diagram chasing calculation, ``$\therefore(surj)$" reads as:
%given a (valid) diagram, add label $(surj)$ to the corresponding arrow.\newline
(a)
The definition of a lifting property $f\rttt g$ {\em up to conjugation}.
%: for each $i:A\lra X$ and $j:B\lra Y$
%making the square commutative, i.e.~$f\circ j=i\circ g$, there is a diagonal arrow $\tilde j:B\lra X$ making the total diagram
%$A\xra f B\xra {\tilde j} X\xra g Y, A\xra i X, B\xra j Y$ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$.
%We say that $f$ lifts wrt $g$, $f$ left-lifts wrt $g$, or $g$ right-lifts wrt $f$.
\newline
(b) A corollary of Sylow theorem: any $p$-subgroup is contained in the Sylow subgroup $S_p$ up to conjugation.
To see this, take $P_1$ to be the trivial group, and note that $P_2$ in ${(0\lra\ZpZ)^{\rttt lr}}$ means $P_2$ is a $p$-group.
To see that this property holds for the Sylow subgroup, note that $P_1\lra P_2$ in ${(0\lra\ZpZ)^{\rttt lr}}$ implies
there is a subgroup subgroup series with $\ZpZ$ quotients connecting $P_1$ and $P_2$, hence
$Card\, P_2/Card\, P_1$ is a power of $p$, hence $Card\,\Imm\,P_2$ is a power of $p$, hence maps to $S_p$
up to conjugation.
% (b) $X\lra Y$ is surjective\newline
% (c) $X\lra Y$ is injective; $X\lra Y$ is an epicmorphism if we forget %never use
%that $\{\bullet\}$ denotes a singleton (rather than an arbitrary object
%and thus $\{\bullet,\bullet\}\lra\{\bullet\}$ denotes an arbitrary morphism $Z\sqcup Z\xra{(id,id)} Z$)\newline
% (d) $X\lra Y$ is injective, in the category of Sets; $\pi_0(X)\lra\pi_0(Y)$ is injective,
% when the diagram is interpreted in the category
%of topological spaces.
\newline (c) The definition of orthogonality with properties/labels.
\newline (d) The diagram expresses that $N$ is self-normalising in $G$. The diagram
says that if $A$ is subnormal in $B$, then any extension of a surjection of $A$ on $N$
to $B$ is necessarily trivial, if the map takes value inside of $G$.
}\end{figure}
\begin{defi}[orthogonal up to conjugation]
Say that two morphisms $A \xra f B$ and $X \xra g Y$ in a category $C$ are {\em orthogonal up to conjugation} and write $f\rttt g$
iff
for each $i:A\lra X$ and $j:B\lra Y$
making the square commutative, i.e.~$f\circ j=i\circ g$, there is a diagonal arrow $\tilde j:B\lra X$
%and an inner automorphism $\sigma:Y\lra Y$ such that %,
and an element $y \in Y$
making the total diagram
%$A\xra f B \xra {\tilde j} X^\sigma \xra g Y, A\xra i X \xra \sigma X^\sigma , B\xra j Y\xra \sigma Y $ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$ (see Figure 3a).
$A\xra f B \xra {\tilde j} X \xra{ygy^{-1}} Y, A\xra i X , B\xra j Y $ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ (ygy^{-1})=j$ (see Figure 3a).
%?????
%?????
Define {\em left/right Quillen negation} or {\em left/right Quillen orthogonal} $P^{\rttt l}$, $P^{\rttt r}$ {\em up to conjugation}
in the obvious way.
\end{defi}
Then the corollary of the Sylow theorem that there is a $p$-subgroup which contains any other $p$-subgroup up to conjugation,
and such a $p$-subgroup is unique up to conjugation, can be expressed as:
each morphism $0\lra G$ decomposes as
$$
0 \xra {(0\lra\ZpZ)^{\rttt lr}} S_p \xra {(0\lra\ZpZ)^{\rttt lrr}} G,
$$
and such decomposition is unique up to conjugation.
\subsection{Normalisers and orthogonality with properties.}
\begin{defi}[orthogonality with properties]
Let $(u)$ and $(d)$ be two classes of morphisms; we think of them, and write them as, labels on arrows.
Say that two morphisms $A \xra f B$ and $X \xra g Y$ in a category $C$ are {\em $(u)\rtt(d)$-orthogonal} and write $f ((u)\rttt(d)) g$
iff
for each $i:A\lra X$ with property $(u)$ and $j:B\lra Y$ with properby $(d)$
making the square commutative, i.e.~$f\circ j=i\circ g$ there is a diagonal arrow $\tilde j:B\lra X$
making the total diagram
$A\xra f B \xra {\tilde j} X \xra g Y, A\xra i X \xra X , B\xra j Y\xra Y $ commutative, i.e.~$f\circ \tilde j=i$ and $\tilde j\circ g=j$ (see Figure 3a).
%?????
Define {\em left/right Quillen negation} or {\em left/right Quillen orthogonal} $P^{((u)\rtt(d)) l}$, $P^{((u)\!\rtt\!(d)) r}$ {\em up to conjugation}
in the obvious way.
\end{defi}
A subgroup $N$ of $G$ is self-normalising iff
$$
N \lra G \text{ is in } \left((0\lra *)^{\rtt lr}\right)^{((surj)\!\!\rtt\!\!(all))r}$$
%Little attempt has been made to go beyond these examples.
%Hence open questions remain: are there other interesting examples of lifting properties in the category of (finite) groups?
%Can a complete group-theoretic argument be reformulated in terms of diagram chasing, say
%the classification of CA-groups or $pq$-groups, or
%elementary properties of subgroup series; can category theory notation
%be used to make expositions easier to read?
%Can these reformulations be used in automatic theorem proving?
%Is there a decidable fragment of (finite) group theory
% based on the Quillen lifting property and, more generally,
% diagram chasing, cf.~[GLS]?
%The definition of the $p$-core as ``the unique largest subnormal $p$-subgroup'' can be described by the decomposition
%$$0 \xra { (0\lra \ZpZ)^{\rttt lr}} S_p \xra (0\lra *)^{\rttt lr} }G $$
%
%
%?????????????
%\noindent
%\subsubsection{\em Discussion.} Sylow theorem implies in a finite group $G$, each $p$-subgroup is contained in a maximal one $Syl_p(G)$,
%$Ord\, G/Ord\, Syl_p(G)$ is prime to $p$,
%and the maximal $p$-subgroups are conjugated by an inner automorphism.
%
%It is not clear how to express this in a satisfactory manner in terms of category theory (diagram chasing).
%Perhaps something along the following lines:
%(in the category of finite groups)
%each arrow $0\lra G$ decomposes as
%$$0\xra{(p\text{-group})} Syl_p(G) \xra{(\text{prime to }p)} G$$
%uniquely up to conjugation.
%Here
%label $A\xra{(p\text{-group})} B $ may mean something like $Ord\, B/Ord\, Im\, A$ is a power of $p$,
%and label $B \xra{(\text{prime to }p)} C$ may mean something like $Ord\, C/Ord\, Im\, B$ is prime to $p$.
%
%%Sylow theory says more: the maximal $p$-subgroups are in fact conjugated.
%We remark that the notion of
%an inner automorphism can be reformulated in a diagram chasing manner.
%An inner automorphism $g\longmapsto a g a^{-1}$ of a group $G$ extends to an automorphism $h\longmapsto \iota(a)h\iota(a)^{-1}$
%of a group $H$ for any embedding $\iota:G\lra H$. [Inn, Sch] show this is a characterisation:
%an automorphism $\sigma :G\lra G$ is inner iff it extends to an automorphism of $H$ for any embedding $\iota:G\longrightarrow H$.
%See [Inn] and references therein for several more similar reformulations.
%
%
%
%
%
%The Feit-Thompson theorem can be expressed as a combination of lifting properties:
%the theorem says says that each (finite) group of odd order is soluble, i.e.
%for each perfect finite group $G$ and each finite group $H$,
%$$
%0\lra {\Bbb Z}/2{\Bbb Z} \rtt 0\lra H \implies 0\lra G \rtt 0\lra H.$$
%
%Note that all these examples but the last one have a flavour of negation---a notion
%being defined by the lifting property
%with respect to the simplest counterexample.
%
%\def\rrt#1#2#3#4#5#6{\xymatrix{ {#1} \ar[r]^{} \ar@{->}[d]_{#2} & {#4} \ar[d]^{#5} \\ {#3} \ar[r] \ar@{-->}[ur]^{}& {#6} }}
\def\rrtt#1#2#3#4#5#6#7{\xymatrix{ {#1} \ar[r]^{} \ar@{->}[d]_{#2} & {#4} \ar[d]_{#5}^{#6} \\ {#3} \ar[r] \ar@{-->}[ur]^{}& {#7} }}
\begin{figure}
\begin{center}
\small
$(a)\ \rrt {0} {} {\Bbb Z} H {\therefore(surjective)} G $%\
%$(b)\ \rrt { \left} {.} {\left} H {\therefore(injective)} G $%\
$(b)\ \rrt {\Bbb Z} {} 0 H {\therefore(injective)} G $%\
\vskip3mm
$(c)\ \rrt { \left} {.} {\left} A {\therefore(abelian)} 0 $%\
%$(d)\ \rrt { \left} {.} {\left} A {\therefore(CA-group)} 0 $\ %
$(d)\ \rrt {G} {\therefore(perfect)} {0} A {(abelian)} 0 $%\ \
$(e)\ \rrt G {(perfect)} {0} {H} {\therefore(soluble)} {0}$\
\vskip3mm
$(f)\ \rrt {{\Bbb Z}/p{\Bbb Z}} {.} {0} {H} {\therefore(\,p\,\nmid\,\textrm{card}H)} {0}$
%$(f')\ \rrt {{\Bbb Z}/2{\Bbb Z}} {.} {0} {H} {\therefore(odd)} {0}$
$(g)\ \rrt {H} {(\,p\,\nmid\,\textrm{card}H)} {0} {G} {\therefore(\textrm{card}H=p^n\text{ for some }n)} {0}$\\
\vskip3mm
$(h)\ \rrt {{\Bbb Z}/2{\Bbb Z}} {.} {0} {H} {\therefore(odd)} {0}$
$(i)\ \rrt {{\Bbb Z}/2{\Bbb Z}} {.} {0} {H} {\therefore(soluble)} {0}$
$(k)\ \rrt {N} {\therefore(normal\ closure)} {H} {0} { } {G}$
$(l)\ \rrtt {N} {(normal\ closure)} {H} {D} {} { \therefore(subnormal) } {G}$%
$(m)\ \rrtt {N} {(normal\ closure)} {H} {G} {\Delta} { \therefore(nilpotent) } {G\times G}$
\end{center}
\caption{\label{fig5}\footnotesize %tiny %small
Lifting properties/Quillen negations. Dots $\therefore$ indicate free variables.
Recall these diagrams represent rules in a diagram chasing calculation
and ``$\therefore(label)$" reads as:
given a (valid) diagram, add label $(label)$ to the corresponding arrow.
A diagram is valid
iff for every commutative square of solid arrows
with properties indicated by labels,
there is a diagonal (dashed) arrow making the total diagram commutative.
A single dot indicates that the morphism is a constant.\newline
(a) a homomorphism $H\lra G$ is surjective, i.e.~for each $g\in G$ there is $h\in H$ sent to $g$\newline
(b) a homomorphism $H\lra G$ is injective, i.e.~the kernel of $H\lra G$ is the trivial group\newline
(c) a group is abelian iff each morphism from the free group of two generators
factors through its abelianisation ${\Bbb Z}\times {\Bbb Z}$.\newline
% (d) a group is a CA-group iff the centraliser of each element is abelian, equivalently iff the commutavity relation $[x,y]=0$
% is transitive, or iff each morphism from the amalgamated product $ \left$
% factors through its abelianisation ${\Bbb Z}\times {\Bbb Z}\times {\Bbb Z}$.\newline
(d) a group $G$ is perfect, $G=[G,G]$, iff it admits no non-trivial homomorphism to an abelian group\newline
(e) a finite group is soluble iff it admits no non-trivial homomorphism from a perfect group;
more generally, this is true in any category of groups with a good enough dimension theory.\newline
%(d) by Cauchy's theorem, a finite group has an odd number of elements iff it contains no involution $e,e^2=1, e\neq1$
(f) by Cauchy's theorem, a prime $p$ divides the number of elements of a finite group $G$
iff the group contains an element $e,e^p=1, e\neq1$ of order $p$\newline
(f) a group has order $p^n$ for some $n$ iff iff the group contains no element $e,e^l=1, e\neq1$ of order $l$ prime to $p$\newline
(h) by Cauchy's theorem, a finite group has an odd number of elements iff it contains no involution $e,e^2=1, e\neq1$\newline
(i) The Feit-Thompson theorem says that each group of odd order is soluble,~i.e.~it says that this diagram chasing
rule is valid in the category of finite groups. Note that it is not a definition of the label unlike the other
lifting properties.\newline
(k) a group $H$ is the normal closure of the image of $N$ iff $N\lra H \,\rtt 0 \lra G$ for an arbitrary group $G$\newline
(l) $D\lra G$ is injective and the subgroup $D$ is a subnormal subgroup
of a finite group $G$ iff $D \lra G$
right-lifts wrt any map $N\lra H$ such that $H$ is the normal closure of the image of $N$\newline
(m) a group $G$ is nilpotent iff the diagonal map $G\xra \Delta G\times G$, $g\mapsto (g,g)$
right-lifts wrt any inclusion of a subnormal subgroup $N\lra H$
}\end{figure}
\section*{Acknowledgments and historical remarks.}
%It seems embarrassing to thank anyone for ideas so trivial, and
%we do that in the form of historical remarks....
This work is a continuation of [DMG]; early history is given there.
Examples here were motivated by a discussion with S.Kryzhevich.
I thank Paul Schupp for pointing out the characterisation of inner
automorphisms of [Sch]. I thank M.Bays, K.Pimenov, V.Sosnilo and S.Synchuk for proofreading,
and several students for encouraging and helpful discussions.
I thank David Bradley-Williams for pointing out the example of Burnside groups.
Special thanks are due to M.Bays for helpful discussions.
I wish to express my deep thanks to Grigori Mints, to whose memory this paper is dedicated \dots
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\end{document}