# On the realization problem of plane real algebraic curves as Hessian curves

Angelito Camacho Calderón and Adriana Ortiz Rodríguez  Work supported by
CONACyT Work partially supported by DGAPA-UNAM grant PAPIIT-IN108112 and
N103010

###### Abstract

The Hessian Topology is a subject having interesting relations with several
areas, for instance, differential geometry, implicit differential equations,
analysis and singularity theory. In this article we study the problem of
realization of a real plane curve as the Hessian curve of a smooth function.
The plane curves we consider are constituted either by only outer ovals or
inner ovals. We prove that some of such curves are realizable as Hessian
curves.

Keywords: Hessian curves, Hessian Topology, Real algebraic curves

MS classification: 53A15; 53A05.

## Introduction

It is well known that the points of a generic smooth surface in
$\mathbb{R}^{3}$ ($\mathbb{R}$P3) are classified as elliptic, parabolic and
hyperbolic. The geometric structure of such a surface is described by the
behavior of the sets conformed by these points, that is, the set of parabolic
points is a smooth curve called parabolic curve whose complement is a disjoint
union of elliptic and hyperbolic domains. A problem that has been studied for
a long time is the description of this geometric structure, in particular the
topology of the parabolic curve [13], [9], [5], [2], [3], among others. When
the surface is expressed locally as the graph $\,z=f(x,y)\,$ of a smooth
function $f$, the parabolic curve is the image under $f$, of the Hessian curve
of $f$, which is the set of zeros of the Hessian polynomial of $f$, Hess
$f:=f_{xx}f_{yy}-f_{xy}^{2}$. Moreover, if the surface is the graph of a
smooth function defined on the whole plane, then this local description is
global, and so, to understand the behavior of the parabolic curve in this
case, it is enough to study the Hessian curve.

Some topics concerning the Hessian Topology [1], [4] (problems 2000-1, 2000-2,
2001-1, 2002-1) are based in the study of the following problems. How many
compact connected components can have the (smooth) parabolic curve of the
graph of a real polynomial of degree $n$ in two variables? and, how are they
distributed? [10], [11], [6], [7], [8]. Another such a problem is to determine
the number of connected components of the parabolic curve of a smooth
algebraic surface of fixed degree $n$ in the real projective space [14], [10],
[6]. The realization problem of a smooth plane curve as the Hessian curve of a
smooth function is another such a problem of this subject, which is treated
here. The meaning of all these problems is to find out if the property of
being Hessian curve imposes conditions on the topology and geometry of curves
and surfaces.

A connected component of a compact smooth plane curve is called outer oval if
it is contained in the outside component of the complement of each of the
connected components of the curve. In this paper we give a family of real
polynomials $\\{f_{k}\\}_{k\geq 2}\,$ such that the Hessian curve of $f_{k}$
consists exactly of $k-2$ outer ovals. Jointly, in Theorem 2, we give a
detailed description of the geometric structure of the graph of each one of
these polynomials.

In Theorem 10, for each even natural number $k$, we present real polynomials
of degree $\,k\,$ such that its Hessian curve is formed exactly by $k-2$
concentric circles. We also describe the geometric structure of the graph of
these polynomials. In an analogue form, we contribute to the odd case with a
family of real functions $\\{f_{r}\\}_{r\geq 1}\,$ such that the Hessian curve
of $f_{r}$ are constituted by $2r-1$ concentric circles, Theorem 11.

To conclude, we show in Proposition 12, that if a smooth plane curve $\,C\,$
is equivalent by an affine transformation on the plane to any of the previous
Hessian curves, then $C$ is also a Hessian curve.

## Preliminaries

Points on a smooth surface (not necessarily generic) in $\mathbb{R}^{3}$
($\mathbb{R}$P3) can be classified in terms of the maximal order of contact of
the tangent lines to the surface at each point. A point $p$ of a surface is
elliptic if all lines tangent to the surface at $p$ have a contact of order
$2$ with the surface at that point. The tangent lines having an order of
contact at least three with the surface are known as asymptotic lines. A point
$p$ is hyperbolic if it has exactly two asymptotic lines. We say that an
hyperbolic point is an inflexion point if it has at least one asymptotic line
with order of contact greater than $3$.

The points on the surface that are not elliptic or hyperbolic will be called
parabolic points and the set constituted by them will be referred to as
parabolic curve. This curve is formed by the following types of points (some
of them may not appear). A generic parabolic point is a point which has
exactly one asymptotic line and the order of contact of this line with the
surface is $3$. A special parabolic point (other authors call it Gaussian cusp
or godron) is a point at which the parabolic curve is smooth locally, the
4-jet (without constant neither linear terms) of the function defining the
surface at this point is not a perfect square and moreover, it has exactly one
asymptotic line whose order of contact is at least 4. Finally, a degenerate
parabolic point is a parabolic point that is not generic or special. It is
important to mention that all these type of points are invariant under the
action of the affine group on 3-space.

The directions determined on the $xy$-plane of the asymptotic lines under the
projection $\pi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2},\
(x,y,z)\mapsto(x,y)$, are the solutions of the quadratic differential
equation:

$f_{xx}(x,y)dx^{2}+2f_{xy}(x,y)dxdy+f_{yy}(x,y)dy^{2}=0,$

where the quadratic differential form on the left will be referred to as the
second fundamental form of $f$. Its discriminant defined as

$\Delta_{II_{f}}=f_{xy}^{2}-f_{xx}f_{yy},$

allows us to characterize the type of point in the graph of $f$. That is,
$(p,f(p))$ is elliptic, parabolic or hyperbolic if $\Delta_{II_{f}}(p)$ is
negative, zero or positive, respectively.

## Results and Proofs

##  1 Outer ovals

We recall that a connected component of a compact smooth plane curve is called
outer oval if it is contained in the outside component of the complement of
each of the connected components of the curve.

###### Definition 1

A set of polynomials $\,l_{1},\ldots,l_{n}\in\mathbb{R}[x,y]\,$ of degree one
are in good position if for each $\,i=1,\ldots,n,\,$ the straight line
$\,l_{i}(x,y)$= $0$ contains no critical point of the function $\,\prod_{j\neq
i}l_{j}$.

###### Theorem 2

Let $\,m,n\,$ two natural numbers. Consider the polynomial

$\,f(x,y)=\prod_{i=0}^{m}l_{i}(x,y)\prod_{j=0}^{n}{g_{j}}(x,y)$

of degree $\,m+n+2\,$ given as the product of polynomials of degree one,
$l_{0}(x,y)=y-ax,\,{g_{0}}(x,y)=y-bx,\,l_{i}(x,y)=x-a_{i},\,{g_{j}}(x,y)=x-b_{j}$,
with $\,a,b,a_{i},b_{j}\in\mathbb{R},\,i=1,\ldots,m,j=1,\ldots,n,\,a\neq b\,$
and $\,a_{m}<\cdots a_{1}<0<b_{1}<\cdots<b_{n}$. If the set of polynomials
$\,\\{l_{i}(x,y),\,g_{j}(x,y):i=0,\ldots,m,j=0,\ldots,n\\}\,$ are in good
position, then the Hessian curve of $f$ consists exactly of $\,m+n\,$ outer
ovals. Moreover, the graph of $f$ has $\,3(m+n)\,$ special parabolic points
and its inflexion curve is the set $\,\\{f(x,y)=0\\}$.

Proof. In order to prove that the inflexion curve is the set
$\,\\{f(x,y)=0\\}$, we state the following Remark which is a consequence of
the definition of $f(x,y)$.

###### Remark 3

The restriction of $f$ to any straight line $m\in\mathbb{R}^{2}$ is a one
variable real polynomial such that all its inflexion points are non
degenerated.

###### Lemma 4

The inflexion curve of the graph of $f$ is the set $\,\\{f(x,y)=0\\}$.

Proof. Because the order of contact of each straight line $\,l_{i}(x,y)=0,$
$\,i=1,\ldots,n,\,$ with the graph of $f$ is infinite then these lines are
contained in the inflexion curve. Inversely, suppose that $p$ is a point of
the inflexion curve and does not belong to the set $\,L:=\\{f(x,y)=0\\}$. So,
at least one asymptotic direction of $p$ has order of contact equal to or
greater than 4 with the graph of $f$. This implies that the restriction of $f$
to the line, which is the projection to the $xy$-plane of this asymptotic
line, has a degenerated inflexion point. This contradicts Remark 3.

###### Lemma 5

A point is in the intersection of the parabolic curve with the inflexion curve
if and only if this point is a special parabolic point or a degenerated
parabolic point.

Proof. If a point belongs to both curves, parabolic and inflexion, it has an
asymptotic line with order of contact at least 4 with the surface. Then, the
point is special parabolic or degenerated. Inversely, because a special
parabolic point satisfies that its asymptotic line has order of contact at
least 4 with the surface, it lies in the inflexion curve. It only remains to
prove that any degenerated point lies in the inflexion curve. Note that a
degenerated point has one or an infinite number of asymptotic lines. For the
first case, if the Hessian curve is singular at this point, the asymptotic
line has order of contact at least 4 with the surface, so it lies in the
inflexion curve. Remaining in the first case, suppose that the Hessian curve
is smooth at the point and the 4-jet (without constant neither linear terms)
of the function is a perfect square. This implies that the point lies in the
inflexion curve. For the case in which every tangent line at the point is an
asymptotic line, we have that the second fundamental form of $f$ at the point
is identically zero, that is, each tangent line has order of contact at least
4 with the surface.

###### Lemma 6

The intersection of the parabolic curve with the inflexion curve consists of
$\,3(m+n)$ points.

Proof. After some straightforward calculus we have that the restriction of the
Hessian of $f$ to the straight line $\,x=a_{i},\,$ for $i=1,,\ldots,m,\,$ is

$\displaystyle\mathrm{Hess}f_{|_{x=a_{i}}}(x,y)$ $\displaystyle=$
$\displaystyle-(2y-(a+b)a_{i})^{2}\Big{(}\prod_{j=1}^{n}(a_{i}-b_{j})\sum_{k=1}^{m}\prod_{s\neq
k}^{m}(a_{i}-a_{s})\Big{)}^{2}$ $\displaystyle=$
$\displaystyle-(2y-(a+b)a_{i})^{2}\,C_{i}^{2},$

where $\,C_{i}=\prod_{j=1}^{n}(a_{i}-b_{j})\sum_{k=1}^{m}\prod_{s\neq
k}^{m}(a_{i}-a_{s})\,$ is a nonzero real number for each $\,i=1,\ldots,m.$

Then $\mathrm{Hess}f_{|_{x=a_{i}}}(x,y)=0\,$ if and only if
$\,y=\frac{(a+b)a_{i}}{2},\,i=1,\ldots,m.$ The points
$\left(a_{i},\frac{(a+b)a_{i}}{2}\right),$ $i=1,\ldots,m,$ are between the
line $l_{0}(x,y)=0$ and the line $g_{0}(x,y)=0$ because
$\,ba_{i}<\frac{(a+b)a_{i}}{2}<aa_{i}$. Moreover, the other points on the
lines $l_{i}=0,\,i=1,\ldots,m,$ are hyperbolic points.

By a similar analysis for the lines $\,x=b_{j}\,$ for $j=1,,\ldots,n$ we have
that the points $\left(b_{j},\frac{(a+b)b_{j}}{2}\right),$ $j=1,\ldots,n,$ are
parabolic and they are in the interior of the compact segment
$[ab_{j},bb_{j}]$ of the line $\,x=b_{j}\,$.

Now, let us analyze the intersection of the parabolic curve with the straight
lines $l_{0}(x,y)=0$ and $g_{0}(x,y)=0$. Replacing $\,y=ax$ in the Hessian
polynomial of $f$ we obtain

$\displaystyle\mathrm{Hess}f(x,y)|_{y=ax}=-(a-b)^{2}\left[\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\right.\qquad\qquad\qquad$
$\displaystyle\left.+\,x\left(\sum_{l=1}^{m}\prod_{i\neq
l}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})\right)\right]^{2}.$

Note that the expression being inside of the square brackets is the first
derivative of the one variable polynomial $\displaystyle
g(x)=x\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})$ of degree $m+n+1$
which has $m+n+1$ simple zeros and $m+n$ simple critical points. This implies
that $\mathrm{Hess}f(x,y)|_{y=ax}$ has $m+n$ simple real roots.

By an analogous analysis for the straight line $\,y=bx$ we obtain that
$\mathrm{Hess}f(x,y)|_{y=bx}$ has $m+n$ simple real roots.

###### Lemma 7

If the set of polynomials
$\,\\{l_{i}(x,y),\,g_{j}(x,y):i=0,\ldots,m,j=0,\ldots,n\\}\,$ are in good
position, then $\,\\{f(x,y)=0\\}$ contains no degenerated parabolic point.

Before starting with the proof of this Lemma we state the following Remark
which is obtained by a suitable grouping of terms of the Hessian polynomial of
$f$.

###### Remark 8

The Hessian polynomial of $f$ can be written as

$\mathrm{Hess}f(x,y)=\beta(x)\left(y-\frac{(a+b)x}{2}\right)^{2}+\alpha(x),$

where

$\displaystyle\alpha(x)=-(a-b)^{2}\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})\left[\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})\right.\qquad\qquad\quad$
$\displaystyle+2x\left(\sum_{l=1}^{m}\prod_{i\neq
l}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})\right)\qquad\quad\,\,\,$
$\displaystyle+\frac{x^{2}}{2}\left(\sum_{l=1}^{m}\sum_{s\neq
l}^{m}\prod_{i\neq
l,s}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\sum_{k=1}^{n}\sum_{t\neq
k}^{n}\prod_{j\neq k,t}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})\right.$
$\displaystyle\left.\left.+2\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\sum_{l=1}^{m}\prod_{i\neq l}^{m}(x-a_{i})\right)\right],\hskip
150.79968pt$
$\displaystyle\beta(x)=2\left(y-\frac{(a+b)x}{2}\right)^{2}\left[\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})\bigg{(}\sum_{l=1}^{m}\sum_{s\neq
l}^{m}\prod_{i\neq l,s}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})\right.$
$\displaystyle+\sum_{k=1}^{n}\sum_{t\neq k}^{n}\prod_{j\neq
k,t}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})+2\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\sum_{l=1}^{m}\prod_{i\neq l}^{m}(x-a_{i})\bigg{)}\qquad\quad$
$\displaystyle\left.-2\left(\sum_{l=1}^{m}\prod_{i\neq
l}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})\right)\right].\qquad\qquad$

Proof. (Lemma 7). We shall prove that the points lying in both curves,
parabolic and inflexion, are all special points.

The next calculus shows that the second fundamental form of $f$ at each point
$q\in\\{f(x,y)=0\\}\,$ is nonzero. Let suppose that $\,q\in\\{l_{i}(x,y)=0\\}$
for some $i\in\\{0,\ldots,m\\}$ (the following arguments are valid for the
case that $q$ belongs to some $l_{i}(x,y)=0$). So, $\,f(x,y)=l(x,y)l_{i}(x,y)$
where $l(x,y)=\prod_{k=0,k\neq i}^{m}l_{k}(x,y)\prod_{j=0}^{n}g_{j}(x,y).$
Then,

$\displaystyle f_{xx}(q)=2\frac{\partial l_{i}}{\partial x}(q)l_{x}(q),\qquad
f_{yy}(q)=2\frac{\partial l_{i}}{\partial y}(q)l_{y}(q),$ $\displaystyle
f_{xy}(q)=\frac{\partial l_{i}}{\partial x}(q)l_{y}(q)+\frac{\partial
l_{i}}{\partial y}l_{x}(q).\qquad\qquad$

The hypothesis of good position implies that $q$ can not be critical point of
$l$. So, $l_{x}(q)\neq 0$ or $l_{y}(q)\neq 0$ and the second fundamental form
of $f$ at $q$ is non degenerated. That is, each point has exactly one
asymptotic line.

Now, let $p$ be a point in the intersection of the parabolic curve with the
inflexion curve. We shall prove that the Hessian curve at $p$ is non singular.
The proof will be in two cases.

  * •

The point $p$ lies in any of the lines $\,l_{0}(x,y)=0\,$ or $\,g_{0}(x,y)=0$.
After an appropriate affine transformation of the $xy$-plane, we can suppose
that such a line is the line $y=0$. Denote by $\,x_{1},\ldots,x_{m+n+1}$ the
intersection points of the line $y=0$ with all other lines. Since
$f(x,y)=y\prod_{k=1}^{m+n+1}r_{k}(x,y)$, where $\,r_{k}(x,y)$ are real
polynomials of degree one, then,

$f_{y}(x,y)|_{y=0}=\prod_{k=1}^{m+n+1}r_{k}(x,y)|_{y=0}.$

Define $\,f_{y}(x,y)|_{y=0}=F(x)$. The polynomial $F(x)$ is of degree $m+n+1$
and its roots are real and simple because they are the points
$\,x_{1},\ldots,x_{m+n+1}$. Moreover, it has $m+n$ non degenerated critical
points, that is, the function $\,F^{\prime}(x)=f_{xy}(x,y)|_{y=0}\,$ has $m+n$
simple real zeroes: exactly one in the interior of each segment
$\,[x_{i},x_{i+1}],\,i=1,\ldots,m+n$ and $\,f_{xxy}(x,y)|_{y=0}\neq 0\,$ at
these roots. Note that

$(\mathrm{Hess}f(x,y))|_{y=0}=-(f_{xy}(x,y))^{2}|_{y=0}=-(F^{\prime}(x))^{2}.$

So, the point $p$, after applying the affine transformation, has the form
$\tilde{p}=(\tilde{x},0)$, where $\tilde{x}$ is one of the critical points of
$F$. Moreover, $\,f_{xy}(x,y),\,f_{xx}(x,y)$ are zero at this point. Finally,
since the second fundamental form of $f$ at $\tilde{p}$ is non zero and
$\,f_{xxy}(\tilde{p})\neq 0$, then

$\displaystyle\left(\frac{\partial}{\partial
y}\mathrm{Hess}f\right)(\tilde{p})=(f_{xx}f_{yyy}+f_{xxy}f_{yy}-2f_{xy}f_{xyy})|_{\tilde{p}}$
$\displaystyle=f_{xxy}(\tilde{p})f_{yy}(\tilde{p})\neq 0.\hskip 68.00206pt$

  * •

The point $p$ lies in one of the lines $\,l_{1}(x,y)=0,\ldots,l_{m}(x,y)=0,$
$g_{1}(x,y)=0,\ldots,g_{n}(x,y)=0.$ In the proof of Lemma 6 it is shown that
$p$ has the form $\left(x_{0},\frac{(a+b)x_{0}}{2}\right)\,$ where
$\,x_{0}\in\\{a_{1},\ldots,a_{m},b_{1},\ldots,b_{n}\\}$. So, by Remark 8 we
have $\,\frac{\partial}{\partial
y}\mathrm{Hess}f(p)=\alpha^{\prime}(x_{0})\neq 0$.

Note that the asymptotic line at each one of these points is a line of the set
$\,\\{l_{i}(x,y)=0,\,g_{j}(x,y)=0:i=0,\ldots,m,j=0,\ldots,n\\}$ because the
parabolic curve is tangent to this set.

It only remains to prove that the 4-jet (without constant neither linear
terms) of the function at these points is free square. For the points
$\left(a_{i},\frac{(a+b)a_{i}}{2}\right),$ this assertion follows by noting
that the coefficient of the $xy^{2}$ term is nonzero while the coefficient of
$y^{4}$ is zero.

In order to prove the compacity of the parabolic curve consider the bounded
connected components of the complement of $\,\\{f(x,y)=0\\}\,$ in the
$xy$-plane. Take its closure of each one and the union of them. Let us denote
by $W$ the complement of this union.

###### Lemma 9

The points belonging to the set $W$ are hyperbolic.

Proof. We partition the set $W$ in 4 regions which are represented in Figure
1.

Figure 1: Partition of the set $W$ in 4 regions

Consider Region I. It was shown in the proof of Lemma 6 that the points lying
in some of the straight lines $\,l_{i}(x,y)=0,\,{g_{j}}(x,y)=0$, with
$\,i=0,\ldots,m,\,j=0,\ldots,n\,$ of Region I (or II) are hyperbolic. So, it
only remains to show that the points of Region I and belonging to the
complement of $\,\\{f(x,y)=0\\}\,$ are hyperbolic. Let us denote by $q$ such a
point. Consider the straight line $m$ on the $xy$-plane parallel to the
$y$-axis and passing through $q$. The restriction $\,f|_{m}\,$ of $\,f$ to
this line is a one variable polynomial of degree 2 with two real roots (the
intersections with the straight lines $\,l_{0}(x,y)=0,\,g_{0}(x,y)=0$). So,
$f(q)$ and the derivative of $\,f|_{m}\,$ at $q$ are positive or negative.
That is $\,f|_{m}\,$ is convex or concave at $q$, respectively.

Suppose that $f(q)$ is positive. Denote by $t$ the straight line tangent to
the level curve $\,f(x,y)=f(q)$ at $q$. Because the derivative of $\,f|_{t}\,$
is zero, the line $t$ is different from the line $m$. The function
$\,f|_{t}\,$ has a non degenerated critical point at $q$ which is a maximum.
So, the function $\,f|_{t}\,$ is concave at $q$. The two previous conclusions
imply that $(q,f(q))$ is a hyperbolic point. The case $\,f(q)<0\,$ is
analogous.

The analysis for Region II is similar.

Now, we consider Region III which is the unbounded connected component
determined by the lines $\,x=b_{n},\,y=ax\,$ and $\,y=bx$. Let $q$ be a point
in such a set. Consider the straight lines $\,m,l\,$ parallels to the lines
$g_{0}(x,y)=0,\,l_{0}(x,y)=0$, respectively, and passing through $q$. The
lines $\,m,\,l$ divide the $xy$-plane in four sectors, one of which, denoted
by $A$, is totally contained in Region III. Take a straight line $t$ such that
it passes through $q$, it divides the sector $A$ in two unbounded sectors and
it is not parallel to any line of $L$. So, the roots of $\,f|_{t}\,$ are in
the left side of $q$.

If $\,f(q)>0$, then the derivative of $\,f|_{t}\,$ at $q$ is positive. So
$\,f|_{t}\,$ is convex at $q$. On the other hand, consider the straight line
passing through $q$ which is parallel to the $y$-axis. Denote it by $r$. So,
$\,f|_{r}\,$ is concave at $q$ because it is a one variable polynomial of
degree two and $\,f(q)>0$. Then, the point $(q,f(q))$ is hyperbolic. The case
for Region IV is similar.

Since there is one special parabolic point at each compact segment of the
straight lines $g_{0}(x,y)=0,\,l_{0}(x,y)=0$, there is at least one connected
component of the Hessian curve on each compact polygone delimited by the lines
$g_{j}(x,y)=0,\,l_{i}(x,y)=0,\,i=0,\ldots,m,\,j=0,\ldots,n.$ Because there are
$m+n$ such as compact polygons, then there are at least $m+n$ connected
components of the Hessian curve. Now, the goal is to prove that the Hessian
curve has exactly $m+n$ connected components. In fact, we shall show that the
Hessian curve has at most $2(m+n)$ vertical tangent lines since this implies
that the Hessian curve has at most $m+n$ connected components.

Let us prove it. Consider the expression of the Hessian polynomial of $f$
given in Remark 8. The parabolic points having a vertical tangent line to the
parabolic curve are given by

$\Big{\\{}\frac{\partial}{\partial
y}\mbox{Hess}f(x,y)=0\Big{\\}}\cap\Big{\\{}\mbox{Hess}f(x,y)=0\Big{\\}}.$

Note that if $x_{0}$ is a real root of the polynomial $\beta(x)$ then
$\alpha(x_{0})\neq 0$ because in another case the points of the line $x=x_{0}$
would be parabolic and it contradicts Lemma 9. So, the parabolic points having
a vertical tangent line are the points
$\,\left(\tilde{x_{0}},\frac{(a+b)\tilde{x_{0}}}{2}\right)\,$ such that
$\tilde{x_{0}}$ is a real root of $\alpha(x)$. The proof concludes by noting
that $\alpha(x)$ is a one variable real polynomial of degree $2(m+n)$.

##  2 Concentric circles. Even case

###### Theorem 10

Consider the real polynomial of degree $\,2n$

$\,f(x,y)=\prod_{i=1}^{n}(x^{2}+y^{2}-m_{i}^{2})\,$

with $\,0<m_{1}<m_{2}<\cdots<m_{n},\,$ $\,m_{i}\in\mathbb{R},i=1,\ldots,n$.
Then, its Hessian curve is non singular and it is formed by $2(n-1)$
concentric circles. Moreover, the unbounded connected component of the
complement of the Hessian curve on the $xy$-plane is elliptic.

Proof. After a straightforward calculus the Hessian polynomial of $f$ is

$\mathrm{Hess}(f)=4\,s(x,y)\,t(x,y),\quad\mbox{where}$ $\displaystyle
s(x,y)=\sum_{j=1}^{n}\prod_{i\neq j}^{n}(x^{2}+y^{2}-m_{i}^{2})\,\quad\mbox{
and}\hskip 82.51282pt$ $\displaystyle t(x,y)=\sum_{j=1}^{n}\prod_{i\neq
j}^{n}(x^{2}+y^{2}-m_{i}^{2})+2(x^{2}+y^{2})\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\prod_{i\neq j,l}^{n}(x^{2}+y^{2}-m_{i}^{2}).$

Note that the graph of the Hessian of $f$, $\,z-\mathrm{Hess}f(x,y)=0,$ is the
rotation about the $z$-axis of the graph of the function
$\,4\tilde{s}(x)\tilde{t}(x),\,$ where

$\widetilde{s}(x)=\sum_{j=1}^{n}\prod_{i\neq j}^{n}(x^{2}-m_{i}^{2})\,\mbox{
and}\qquad$ $\widetilde{t}(x)=\sum_{j=1}^{n}\prod_{i\neq
j}^{n}(x^{2}-m_{i}^{2})+2(x^{2})\sum_{j=1}^{n}\sum_{l\neq j}^{n}\prod_{i\neq
j,l}^{n}(x^{2}-m_{i}^{2}).$

So, the solution curves of $\,s(x,y)=0$ $(\,t(x,y)=0)\,$ correspond to the non
negative real roots of $\widetilde{s}(x)$ $(\,\widetilde{t}(x))$. We shall
prove now that $\widetilde{s}(x)$ and $\widetilde{t}(x)$ have $n-1$ simple
positive real roots, each one. Define the one variable real polynomial

$\widetilde{f}(x):=\prod_{i=1}^{n}(x^{2}-m_{i}^{2}).$

The degree of this polynomial is $\,2n$, it has $2n$ non zero simple roots,
$2n-1$ non degenerated critical points and $2n-2$ simple inflexion points.
Because

$\widetilde{f}_{x}(x)=2x\sum_{j=1}^{n}\prod_{i\neq
j}^{n}(x^{2}-m_{i}^{2})=2x\widetilde{s}(x),$

then $\,\widetilde{s}(x)$ has $2n-2$ non zero simple real roots from which,
$n-1$ are positive by symmetry. Moreover, since

$\widetilde{f}_{xx}(x)=2\sum_{j=1}^{n}\prod_{i\neq
j}^{n}(x^{2}-m_{i}^{2})+4x^{2}\sum_{j=1}^{n}\sum_{l\neq j}^{n}\prod_{i\neq
j,l}^{n}(x^{2}-m_{i}^{2})=\widetilde{t}(x),$

then $\widetilde{t}(x)$ has $2n-2$ simple real roots (different from the roots
of $\widetilde{s}(x)$). Because $\widetilde{f}$ is symmetric with respect to
the origin, then $\widetilde{t}(x)$ has $n-1$ positive roots. Moreover, the
unbounded connected component of the complement of the Hessian curve on the
$xy$-plane is elliptic because the functions $\tilde{s}$ and $\tilde{t}$ are
positive if $\,x>m_{n}.$

##  3 Concentric circles. Odd case

###### Theorem 11

Consider the real valued differentiable function

$\,f(x,y)=\prod_{k=1}^{n}\frac{x^{2}+y^{2}-k^{2}}{x^{2}+y^{2}+1}.$

Its Hessian curve is non singular and it is formed by $\,2n-1\,$ concentric
circles. Moreover, the unbounded connected component of the complement of the
Hessian curve on the $xy$-plane is hyperbolic.

Proof. After a straightforward computation the Hessian polynomial of $f$ is

$\mathrm{Hess}f(x,y)=\frac{4\,s(x,y)\,t(x,y)}{(x^{2}+y^{2}+1)^{2n+3}},\,\,\mbox{
where}$ $\displaystyle s(x,y):=\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{i\neq
j}^{n}(x^{2}+y^{2}-i^{2})\bigg{)},\hskip 110.96556pt$ $\displaystyle
t(x,y):=(1-3x^{2}-3y^{2})\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{i\neq
j}^{n}(x^{2}+y^{2}-i^{2})\bigg{)}+\hskip 28.45274pt$
$\displaystyle\qquad\qquad+2(x^{2}+y^{2})\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\sum_{l\neq
j}^{n}\bigg{(}(l^{2}+1)\prod_{k\neq
j,l}^{n}(x^{2}+y^{2}-k^{2})\bigg{)}\bigg{)}.$

Note that the graph of the Hessian polynomial of $f$, is the rotation about
the $z$-axis of the graph of the function
$\,\frac{4\tilde{s}(x)\tilde{t}(x)}{(x^{2}+1)^{2n+3}}$, where

$\displaystyle\tilde{s}(x)=\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{k\neq
j}^{n}(x^{2}-k^{2})\bigg{)},\,\,\quad\quad\tilde{t}(x)=(1-3x^{2})\sum_{j=1}^{n}\bigg{(}(j^{2}+$
$\displaystyle+1)\prod_{k\neq
j}^{n}(x^{2}-k^{2})\bigg{)}+2x^{2}\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\sum_{l\neq
j}^{n}\bigg{(}(l^{2}+1)\prod_{k\neq j,l}^{n}(x^{2}-k^{2})\bigg{)}\bigg{)}.$

We analyze now the zeros of $\,\tilde{s}(x)\,$ and $\,\tilde{t}(x)$. Consider
the differentiable function $\,\widetilde{f}:\mathbb{R}\rightarrow\mathbb{R}$

$\widetilde{f}(x)=\prod_{k=1}^{n}\frac{x^{2}-k^{2}}{x^{2}+1}.$

This function has exactly $2n$ simple zeros from which $n$ are positive.
Moreover,

$\displaystyle\widetilde{f}_{x}(x)=\frac{2x}{(x^{2}+1)^{n+1}}\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{i\neq
j}^{n}(x^{2}-i^{2})\bigg{)}=\frac{2x\,\widetilde{s}(x)}{(x^{2}+1)^{n+1}},\quad\qquad$
$\displaystyle\widetilde{f}_{xx}(x)=\frac{2}{(x^{2}+1)^{n+2}}\Bigg{[}(1-3x^{2})\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{i\neq
j}^{n}(x^{2}-i^{2})\bigg{)}+\qquad\qquad$
$\displaystyle+2x^{2}\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\sum_{l\neq
j}^{n}\bigg{(}(l^{2}+1)\prod_{k\neq
j,l}^{n}(x^{2}-k^{2})\bigg{)}\bigg{)}\Bigg{]}=\frac{2\,\widetilde{t}(x)}{(x^{2}+1)^{n+2}}.$

Since $\widetilde{f}(x)$ has at least $2n-1$ critical points and the degree of
$\tilde{s}(x)$ is $2n-2$ then, the polynomial $\tilde{s}(x)$ has exactly
$2n-2$ simple roots different from zero ($n-1$ are positive). So, the set
$\,s(x,y)=0\,$ is a disjoint union of $n-1$ concentric circles.

Because the critical points of $\widetilde{f}(x)$ are non degenerated and it
is symmetric respect to the origin, then $\widetilde{f}(x)$ has at least $n-1$
positive inflexion points in the interval $(-n,n),$ exactly one point between
two consecutive critical points. That is, $\widetilde{t}(x)$ has $n-1$
positive real roots in the interval $[0,n)$.

Denote by $\,x_{0}$ the critical point of $\,\tilde{f}(x)$ lying in the
interval $(n-1,n).$ Note that the sign of
$\,\tilde{s}(n-1)=\left((n-1)^{2}+1\right)\prod_{k\neq
n-1}^{n}\left((n-1)^{2}-k^{2}\right)\,$ is negative while the sign of
$\,\tilde{s}(n)=(n^{2}+1)\prod_{i=1}^{n-1}(n^{2}-i^{2})\,$ is positive. From
these remarks and because the function $\,\widetilde{f}$ has exactly one
critical point at each interval $\,(i,i+1),\,i=1,\ldots,n-1$, we have that
$\,x_{0}$ is a minimum. Moreover, $\tilde{t}(x_{0})$ is positive. After a
straightforward calculation, it can be seen that $\,\tilde{t}(n^{2})\,$ is
negative. So, by the main value Theorem, there exists a real number
$\,c\in(x_{0},n^{2})\,$ such that $\tilde{t}(c)=0.$ It implies that
$\tilde{t}(x)$ has $n$ positive real roots. That is, the set $\,t(x,y)=0\,$ is
a disjoint union of $n$ concentric circles different from the circles
described by the equation $\,s(x,y)=0.$ We conclude that the Hessian curve of
$\,f\,$ consists of $2n-1$ concentric circles.

###### Proposition 12

If a smooth plane curve $\,C\,$ is equivalent by an affine transformation of
the plane, to a Hessian curve, then $\,C\,$ is also a Hessian curve.

Proof. Suppose that the curve $C$ is defined by $\,g(x,y)=0,\,$ where $g$ is a
smooth real valued function defined on the plane. By hypothesis, the curve $C$
is equivalent by an affine transformation to a Hessian curve, that is,
$\,g(x,y)=(\mbox{Hess }f)\circ T(x,y),$ where
$T:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is an affine transformation of the
plane whose determinant (of the linear part) is denoted by $J$. After a
straightforward calculation, we have that

$\mbox{Hess }\left(\frac{f\circ T(x,y)}{J}\right)=(\mbox{Hess }f)\circ
T(x,y).$

So, the curve $C$ is also a Hessian curve.

## References

  * [1] Arnold V.I., Astroidal geometry of hypocycloides and the Hessian topology of hyperbolic polynomials, Russ. Math. Surv., (2001), 56, no. 6, 1019-1083.
  * [2] Arnold V. I., Remarks on Parabolic Curves on Surfaces and the Higher-Dimensional Mobius-Sturm Theory, Funct. Anal. Appl. 31 No. 4 (1997), 227-239.
  * [3] Arnold V. I., On the problem of realization of a given Gaussian curvature function, Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 199-206.
  * [4] Arnold V. I., Arnold’s Problems, Springer-Verlag, 2004.
  * [5] Banchoff T., Gaffney T. and Mc.Crory C., Cusps of Gauss Mappings, Res. Notes Math., Pitman 1982.
  * [6] Bertrand B. $\&$ Brugallé E., On the number of connected components of the parabolic curve, C. R. Math. Acad. Sci. Paris 348 (2010), no. 5-6, 287–289.
  * [7] Hernández Martínez L. I., Ortiz-Rodríguez A. $\&$ Sánchez-Bringas F., On the affine geometry of the graph of a real polynomial, J. Dyn. Control Syst., 18, No.4 (2012), 455-465.
  * [8] Hernández Martínez L. I., Ortiz Rodríguez A. $\&$ Sánchez Bringas F., On the Hessian geometry of a real polynomial hyperbolic near infinity. Adv. Geom. 13, No.2 (2013), 277-292.
  * [9] Kergosien, Y.L.; Thom, R. Sur les points paraboliques des surfaces, C. R. Acad. Sci. Paris, Série A, t. 290 (1980), 705-710.
  * [10] Ortiz Rodríguez A., Quelques aspects sur la géométrie des surfaces algébriques réelles, Bull. Sci. Math., 127, No.2, (2003), 149 - 177.
  * [11] Ortiz Rodríguez A., Sottile F., Real Hessian Curves, Bol. Soc. Mat. Mexicana (3), 13, no.1 (2007), 157-166.
  * [12] 18, No.4 (2012), 455-465 Platonova O. A., Singularities of the mutual of a surface and a line, Russ. Math. Surv. 36 (1981), 248-249.
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Adriana Ortiz Rodríguez, Instituto de Matemáticas, Universidad Nacional
Autónoma de México, Area de la Inv. Cient., Circuito Exterior, C. U., México
D.F 04510, México.
e-mail<EMAIL_ADDRESS>
Angelito Camacho Calderón, Instituto de Matemáticas, unidad Cuernavaca,
Universidad Nacional Autónoma de México, Av. Universidad s/n. Col. Lomas de
Chamilpa, c.p.62210, Cuernavaca, Morelos.
e-mail<EMAIL_ADDRESS>


# Embarrassingly Simple Unsupervised Aspect Based Sentiment Tuple Extraction

Kevin Scaria1∗ Abyn Scaria2♠ Ben Scaria†
1Arizona State University 2Georgia Institute of Technology
<EMAIL_ADDRESS>
###### Abstract

Aspect Based Sentiment Analysis (ABSA) tasks involve the extraction of fine-
grained sentiment tuples from sentences, aiming to discern the author’s
opinions. Conventional methodologies predominantly rely on supervised
approaches; however, the efficacy of such methods diminishes in low-resource
domains lacking labeled datasets since they often lack the ability to
generalize across domains. To address this challenge, we propose a simple and
novel unsupervised approach to extract opinion terms and the corresponding
sentiment polarity for aspect terms in a sentence. Our experimental
evaluations, conducted on four benchmark datasets, demonstrate compelling
performance to extract the aspect oriented opinion words as well as assigning
sentiment polarity. Additionally, unsupervised approaches for opinion word
mining have not been explored and our work establishes a benchmark for the
same 111Experiments and results are available at
https://anonymous.4open.science/r/AutoABSA .

##  1 Introduction

Existing approaches for aspect based sentiment analysis (ABSA) subtasks
predominantly fall into two categories: supervised and unsupervised
methodologies. Recent trends in supervised approaches have seen the
formulation of aspect term sentiment classification tasks as an entailment
problem Seoh et al. (2021), machine reading comprehension (MRC) problem Chen
et al. (2021), grid-tagging and table-filling schemes Wu et al. (2020); Zhai
et al. (2023), leveraging techniques such as graph convolution networks Li et
al. (2021b); Chen et al. (2022); Liang et al. (2022), generative approaches
Yan et al. (2021); Zhang et al. (2021), prompt tuning Li et al. (2021a),
instruction tuning Varia et al. (2023); Scaria et al. (2023), multi-task
learning Wang et al. (2022), and supervised contrastive learning Peper and
Wang (2022), which inherently rely on human-labelled annotated datasets.

Figure 1: Approach overview for ATSC subtask

Furthermore, approaches employing large language models (LLMs) also face
challenges in adapting to new domains and necessitate labeled data for fine-
tuning.

Figure 2: An overview of our approach for AOOE subtask

In practice, the curation of annotated datasets proves to be costly and time-
consuming. Numerous unsupervised methodologies have been proposed for aspect
term extraction, employing techniques such as Latent Dirichlet Allocation
(LDA) Das and Chakraborty (2019); Venugopalan and Gupta (2022), topic modeling
Pathan and Prakash (2021), and hybrid candidate filtering Mai and Zhang
(2020). Similarly, for aspect category extraction, several unsupervised
approaches have been introduced, leveraging contrastive attention Tulkens and
van Cranenburgh (2020) and attention-based aspect extraction He et al. (2017).
However, there have been very limited unsupervised approaches proposed for
other complex ABSA subtasks such as aspect oriented opinion (AOOE) extraction
and aspect term sentiment classification (ATSC).

We present a simple unsupervised method for the AOOE and ATSC subtasks that
only necessitates a Part-of-Speech (POS) tagger and domain-adapted word
embeddings trained on the relevant dataset. To the best of our knowledge, our
work is the first to establish a benchmark for the aforementioned ABSA
subtasks in an unsupervised setting. We carry out extensive experiments on
several datasets and the results demonstrate the effectiveness of our approach
in extracting meaningful opinion words and assigning sentiment polarity to
aspects. Additionally, availability of labelled instances improves the
performance by $\sim 1.8\%$ across subtasks. Notably, further analysis reveals
that scaling up the model size improves the performance by $\sim 2\%$ across
subtasks. Furthermore, our findings indicate evidence of domain
generalizability.

##  2 Task Defintion

Consider the $i^{th}$ review sentence, denoted as $S_{i}$, in the training
sample, where $S_{i}=\\{w_{i}^{1},w_{i}^{2},...,w_{i}^{n}\\}$ and $n$ is the
number of tokens in the sentence. Each $S_{i}$ contains a set of aspect terms,
represented by $A_{i}=\\{a_{i}^{1},a_{i}^{2},...,a_{i}^{m}\\}$ with $m\leq n$.
The corresponding opinion terms and sentiment polarities for each aspect term
are denoted by $O_{i}=\\{o_{i}^{1},o_{i}^{2},...,o_{i}^{m}\\}$ and
$SP_{i}=\\{sp_{i}^{1},sp_{i}^{2},...,sp_{i}^{m}\\}$, where
$sp_{i}^{k}\in\\{positive,negative,neutral\\}$.

The system takes both the sentence $S_{i}$ and the aspect term $a_{i}^{k}$ as
the input. For the AOPE subtask, the opinion word $o_{i}^{k}$ corresponding to
the aspect term is extracted whereas for the ATSC subtask, the corresponding
sentiment polarity $sp_{i}^{k}$ is assigned. The Aspect Oriented Opinion
Sentiment Pair Extraction (AOOSPE) subtask aims to extract the opinion-
sentiment tuple $(o_{i}^{k},sp_{i}^{k})$ for each aspect term in a given
sentence.

##  3 Method

###  3.1 Domain Adaptation

In our proposed method, the first step is domain adaptation. To acquire in-
domain embeddings, we conduct pre-finetuning of a language model $LM$ using an
in-domain dataset.

###  3.2 Compound Phrase Extraction

Previous works Hu and Liu (2004); Mai and Zhang (2020); Tulkens and van
Cranenburgh (2020) use POS taggers and dependency trees to extract NOUN chunks
for extracting aspect terms. However, in the second step of our pipeline, we
extract opinion words by identifying compound phrases. This involves
extracting nouns that are modified by sentiment-bearing adjectives. This
opinion relation enables the association of an opinion word with the aspect
term, and subsequently in the assignment of sentiment polarity. For each
$S_{i}$, we investigate the presence of five opinion relations:
"Adj-{mod}-Noun", "Adv-{mod}-Adj", "Adj-{mod}-Noun-{mod}-Noun", and
"Adp-{mod}-Noun".

###  3.3 Attention Weighting

The opinion words extracted using the opinion relations have very poor
precision Xu et al. (2013), owing to false opinion relations, false opinion
targets, and scenarios where there are long-tail opinion targets. It becomes
challenging to associate a candidate opinion word with an aspect term and
simultaneously filter out non-opinion words. Thus as part of this step, we
weigh the candidate opinion terms using the self-attention scores
corresponding to the aspect term. The self-attention matrix for the review
sentence $S_{i}$ with $n$ tokens from a given layer $N_{\text{layer}}$ of the
model $LM$ is obtained using the scaled dot-product attention.

$AttScore_{[a_{i}^{k}]}=\text{softmax}\left(\frac{Q_{[a_{i}^{k}]}\cdot
K^{T}}{\sqrt{d_{k}}}\right)$

Where $a_{i}^{k}$ is the $k^{th}$ aspect term and $Q$, $K$ is the query and
key matrices of the $n$ tokens for the review sentence $S_{i}$. $d_{k}$ is the
dimensionality of the key matrix. The candidate opinion word corresponding to
the index of the maximum attention score w.r.t the aspect term is finally
selected as the opinion word.

$o_{i}^{k}=\text{argmax}\left(AttScore_{[a_{i}^{k}]}\cdot I\right)$

The indicator function $I$ is defined as follows:

$I(w_{i}^{k})=\begin{cases}1,&\text{if }w_{i}^{k}\text{ is a candidate opinion
word}\\\ 0,&\text{otherwise}\end{cases}$

Since transformer models, encode the linguistic syntactic structure in the
lower layers Jawahar et al. (2019), we use attention scores from the first
four layers of the $LM$ model. For each $N_{\text{layer}}$, we aggregate the
attention scores across the $N_{\text{heads}}$ attention heads.

$o_{i}^{k}=\text{argmax}\left(\frac{1}{N_{\text{heads}}}\sum_{N_{\text{heads}}}\left(AttScore_{[a_{i}^{k}]}\right)\cdot
I\right)$

###  3.4 Semantic Polarity Assignment

In this final step, we generate representations $h_{op}$ using the model $LM$
for the opinion terms extracted in the previous step. To assign the semantic
polarity to each opinion term, we compute the cosine similarity between
$h_{o}$ and the polarity label vectors.

$sp_{i}^{k}=\operatorname*{argmax}_{c\in
C}\left(cos(h_{op}\cdot\vec{c})\right)$

Where, $C$ is the set of labels used to assign the sentiment polarity labels
i.e. $\\{positive,negative,neutral\\}$.

##  4 Results and Analysis

###  4.1 Experimental Setup

We utilize the SemEval 2014, 2015, and 2016 datasets for our experimentation.
These datasets serve as benchmarks for ABSA tasks and comprise customer
reviews from three domains: laptops (L14), hotels (H15), and restaurants (R14,
R15, and R16). Following previous work we report the performance only on the
laptops and restaurants dataset. All the results reported are the average
values from 5 runs of the domain adaptation step for each experiment. Further
details and results regarding experiments on other MLMs and auto-regressive
models are available in §B, LABEL:sec:dataset & C.

###  4.2 Sub Task Results

Table 1, 2, and 3 denotes the results of AOOE, ATSC, and AOOSPE subtasks
respectively.

Model | L14 | R14 | R15 | R16
---|---|---|---|---
ELECTRA | 51.98 | 62.84 | 65.32 | 66.67
DeBERTa | 50.18 | 58.41 | 62.32 | 65.11
T5 | 39.98 | 40.18 | 45.65 | 46.67
Pythia | 47.76 | 60.12 | 61.32 | 60.21

Table 1: Results of the AOOE reporting the accuracy.

For the AOOE subtask (refer to Table 1), the ELECTRA model has consistently
achieved the highest scores across all four datasets, outperforming other
models. Following ELECTRA, Pythia, a decoder-only model, has also shown
notable performance, while the seq2seq T5 model has consistently yielded lower
scores. Furthermore, encoder-only models exhibit $\sim 3.58\%$ better
performance compared to decoder-only models. This trend can be attributed to
the encoder models’ capability to better capture the contextual information
necessary for extracting opinion words."

Model | L14 | R14 | R15 | R16
---|---|---|---|---
ELECTRA | 54.34 | 62.13 | 61.78 | 61.93
DeBERTa | 52.21 | 60.34 | 61.43 | 63.23
T5 | 53.69 | 57.30 | 56.61 | 57.22
Pythia | 52.26 | 60.13 | 61.21 | 61.84

Table 2: Results of the ATSC reporting the accuracy.

In the ATSC subtask (refer to Table 2), the ELECTRA model consistently
outperforms other models on the L14, R14, and R15 datasets. However, on the
R16 dataset, the DeBERTa model achieved the highest accuracy score.

Model | L14 | R14 | R15 | R16
---|---|---|---|---
ELECTRA | 51.76 | 61.39 | 61.25 | 60.95
DeBERTa | 49.56 | 57.12 | 60.34 | 62.77
T5 | 34.54 | 39.31 | 43.69 | 44.17
Pythia | 45.12 | 58.33 | 60.17 | 59.25

Table 3: Results of the AOOSPE reporting the accuracy.

Similarly for the AOOSPE subtask (Table 3), the ELECTRA model performs best
for the L14, R14 and R15 dataset, whereas for the R16 dataset, DeBERTa
outperforms other models.

Model | Cross Domain | Join Domain
---|---|---
AOOE | ATSC | AOOSTE | AOOE | ATSC | AOOSTE
L14 | R14 | L14 | R14 | L14 | R14 | L14 | R14 | L14 | R14 | L14 | R14
ELECTRA | 50.79 | 59.11 | 49.99 | 59.62 | 49.70 | 56.76 | 53.43 | 63.79 | 55.79 | 63.32 | 52.96 | 62.37
DeBERTa | 48.98 | 54.78 | 47.59 | 59.43 | 47.88 | 54.73 | 50.95 | 58.71 | 52.54 | 61.05 | 50.21 | 58.11
T5 | 38.79 | 38.09 | 49.24 | 51.42 | 32.86 | 33.54 | 41.01 | 40.65 | 53.87 | 57.98 | 35.02 | 40.43
Pythia | 46.82 | 55.99 | 47.89 | 56.26 | 44.23 | 55.23 | 48.31 | 61.11 | 52.43 | 60.93 | 46.55 | 59.65

Table 4: Results of joint and cross domain adaptation results.

Across all three subtask, the ELECTRA model consistently yields higher
performance. This could be due to better contextual representations, acquired
through its replaced token detection objective, as compared to those of other
encoder models. By learning from the entire input sequence rather than just
the masked segments, ELECTRA exhibits significant benefits in the AOOE
subtask, which leverages the linguistic structural information encoded in
lower layers for extracting opinion terms.

###  4.3 Analysis

In this subsection, we analyze our approach on multiple line of enquiries and
carry out ablation studies in §A.

Figure 3: Accuracy across tasks when different % of labelled instances are
available by the ELECTRA model.

#### Availability of labelled instances improves performance:

We inspect the performance of our approach when labelles instances are
available. We model ATSC as a sequence classification task using the best
performing model ELECTRA. We carry out the following scenarios where there are
5%, 10%, 25%, 50%, and 100% of labelled instance availability of sentiment
polarity labels of the training set. From figure 3, it can be seen that upon
using the finetuned models, across all tasks, there is an improvement of $\sim
2.8\%$ on average. The availability of polarity labels enhances the attention
matrix which helps in better improving the extraction of aspect terms.

#### Competitive cross-domain generalization:

In cross domain setting, we perform the domain adaptation step on a train set
from one domain and evaluate on a test set from different domain. From table 4
it can be seen that our approach has comparable ($\sim 90\%$) performance to
in-domain adaptation. Additionally, the cross-domain performance is $\sim 1.1$
higher for encoder-only models as compared to decoder-only models.

#### Joint-domain adaptation improves performance:

In joint domain setting, the train data of the domains (laptops and
restaurants) are combined to perform the domain adaptation step, and
evaluation is done on individual test sets. From table 4 it is evident that
using joint domain adaptation the performance of our approach increases by
$\sim 1.9\%$. Notably, the joint domain adaptation improves the performance of
the AOOSPE task the most. Similar to previous trend, the encoder-only models
achieve higher performance gains ($\sim 0.4\%$ points) as compared to decoder-
only models.

##  5 Conclusion

We proposed a novel unsupervised learning approach to address the two ABSA
subtasks viz. ATSC and AOOE. Our results establish a strong benchmark for
these subtasks in an unsupervised setting but also demonstrate competitive
performance compared to various supervised approaches. Our approach does not
require the presence of labeled datasets, showcasing its adaptability to low-
resource domains. Finally we have made our code publicly available. We hope
that our contributions will stimulate further exploration and advancements in
unsupervised ABSA methodologies.

## Limitations

Our study confines itself to the widely employed SemEval 2014, 15, and 16
datasets, representing a common practice in recent research. To ensure the
broader applicability of our findings, future investigations should expand
this work to encompass other ABSA datasets, thereby validating the
generalizability of our approach. It is noteworthy that our experiments
exclusively utilized the base variants of the models, each ranging from 100 to
200 million parameters. Subsequent studies might explore the impact of
employing even larger models to gain deeper insights into their performance
nuances. Another limitation lies in our focus on the English language,
implying that our conclusions may not seamlessly translate to other languages.
Addressing this, future research endeavors should encompass a diverse set of
languages, incorporating multilingual datasets and other models.

## Ethical Considerations

We recognize the potential presence of inherent biases in the language models
employed, stemming from the characteristics of the pre-training data. Although
our study did not specifically conduct stress testing, our comprehensive
analysis indicates that no discernible additional concerns are associated with
privacy, fairness, bias, or discrimination. By focusing on aspect-based
sentiment analysis, our research contributes directly to the scientific
discourse in a constructive manner. We are optimistic about the positive
impact of our work on the broader scientific community. Our commitment to
responsible AI usage remains unwavering, and we pledge to consistently
prioritize ethical considerations in all our future research endeavors.

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## Appendix

##  Appendix A Ablation Studies

#### Impact of Domain Adaptation:

We investigate the impact of the domain adaptation step. The table 5 presents
the performance of the best performing model ELECTRA with and without the
domain adaptation step. It is evident that domain adaptation improves the
performance across subtasks by $\sim 2\%$ on an average.

Task | L14 | R14 | R15 | R16
---|---|---|---|---
AOOEda | 51.98 | 62.64 | 65.32 | 66.67
AOOEw/oda | 49.12 | 60.32 | 64.11 | 64.21
ATSCda | 54.34 | 62.13 | 61.78 | 61.93
ATSCw/oda | 52.76 | 61.77 | 60.22 | 59.45
AOOSPEda | 51.76 | 61.39 | 61.25 | 60.95
AOOSPEw/oda | 50.54 | 58.76 | 60.21 | 58.89
Table 5: Results with and without domain adaptation results by the best
performing model for each subtask. $da$, $w/oda$ stands for domain adaptation
and without $da$ respectively.

#### Impact of Massive Domain Finetuning:

The yelp dataset Asghar (2016) is a massive dataset of yelp reviews on
restaurants and products. We utilize this dataset as part of the domain
adaptation step to learn the rich latent dependency between aspects and
opinion terms. The table 6 presents the performance of the best performing
model ELECTRA with and without the massive domain finetuning step. Notably,
this improves the performance across subtasks by $\sim 4.2\%$ on an average.

Task | L14 | R14 | R15 | R16
---|---|---|---|---
AOOEmdf | 54.78 | 65.76 | 67.43 | 68.12
ATSCmdf | 57.32 | 64.54 | 63.34 | 63.37
AOOSPEmdf | 59.01 | 64.34 | 64.49 | 64.81
Table 6: Results of massive domain finetuning results across subtask by best
performing model. $mdf$ stands for massive domain finetuning.

##  Appendix B Experimental Setup

#### Models Used:

We experiment with various types of models such as BERT Devlin et al. (2019),
ALBERT Lan et al. (2020), RoBERTa Zhuang et al. (2021), ELECTRA Clark et al.
(2020) and DeBERTa He et al. (2020), T5 Roberts et al. (2019), Pythia Biderman
et al. (2023), GPTNeo, and Cerebras Dey et al. (2023). To ensure a fair
comparison of the model’s performance, we use the base variant of the MLMs and
T5, the 160M parameter model for Pythia, the 125M parameter model for GPT-Neo,
and the 111M parameter model for Cerebras.

#### Hyperparameters

GPU: 1xNvidia Tesla P40, Domain Adaptation Step Batch Size: 16, Gradient
Accumulation Steps: 2, Initial learning rate: 5e-5, Num of Epochs: 5

#### Evaluation Metric:

We report the accuracy scores for the AOOE, ATSC & AOOSPE subtasks.

##  Appendix C Further Results

The following tables 7, 8 and 10, present the experiements run on a variety of
models as mentioned in §B.

Model | L14 | R14 | R15 | R16
---|---|---|---|---
BERT | 49.79 | 59.01 | 61.98 | 63.74
ALBERT | 50.37 | 62.54 | 64.26 | 66.25
RoBERTa | 48.08 | 58.81 | 60.33 | 60.82
ELECTRA | 51.98 | 62.84 | 65.32 | 66.67
DeBERTa | 50.18 | 58.41 | 62.32 | 65.11
T5 | 39.98 | 40.18 | 45.65 | 46.67
GPTNeo | 46.91 | 58.67 | 60.09 | 60.14
Pythia | 47.76 | 60.12 | 61.32 | 60.21
Cerebras | 49.21 | 56.64 | 60.92 | 61.75

Table 7: Results of the AOOE subtask reporting accuracy scores.

Model | L14 | R14 | R15 | R16
---|---|---|---|---
BERT | 52.03 | 60.32 | 60.33 | 60.68
ALBERT | 52.40 | 61.73 | 61.77 | 62.69
RoBERTa | 52.58 | 60.32 | 60.74 | 60.78
ELECTRA | 54.34 | 62.13 | 61.78 | 61.93
DeBERTa | 52.21 | 60.34 | 61.43 | 63.23
T5 | 53.69 | 57.30 | 56.61 | 57.22
GPTNeo | 51.23 | 61.77 | 60.91 | 61.25
Pythia | 52.26 | 60.13 | 61.21 | 61.84
Cerebras | 53.50 | 59.96 | 61.11 | 62.01

Table 8: Results of the ATSC subtask reporting the accuracy scores.

Model | Cross Domain | Join Domain
---|---|---
AOOE | ATSC | AOOSTE | AOOE | ATSC | AOOSTE
L14 | R14 | L14 | R14 | L14 | R14 | L14 | R14 | L14 | R14 | L14 | R14
BERT | 48.84 | 57.87 | 48.01 | 58.72 | 47.59 | 56.32 | 50.21 | 60.03 | 52.88 | 60.77 | 49.43 | 58.37
ALBERT | 47.99 | 58.50 | 47.57 | 57.35 | 46.23 | 55.07 | 50.80 | 62.76 | 52.80 | 61.75 | 49.21 | 61.77
RoBERTa | 46.91 | 56.51 | 48.13 | 55.37 | 44.94 | 54.36 | 48.43 | 59.02 | 53.11 | 60.91 | 47.98 | 58.76
ELECTRA | 50.79 | 59.11 | 49.99 | 59.62 | 49.70 | 56.76 | 53.43 | 63.79 | 55.79 | 63.32 | 52.96 | 62.37
DeBERTa | 48.98 | 54.78 | 47.59 | 59.43 | 47.88 | 54.73 | 50.95 | 58.71 | 52.54 | 61.05 | 50.21 | 58.11
T5 | 38.79 | 38.09 | 49.24 | 51.42 | 32.86 | 33.54 | 41.01 | 40.65 | 53.87 | 57.98 | 35.02 | 40.43
GPTNeo | 48.79 | 56.55 | 46.38 | 58.28 | 45.65 | 56.05 | 50.87 | 58.93 | 52.31 | 62.14 | 47.93 | 58.65
Pythia | 46.82 | 55.99 | 47.89 | 56.26 | 44.23 | 55.23 | 48.31 | 61.11 | 52.43 | 60.93 | 46.55 | 59.65
Cerebras | 46.93 | 53.65 | 48.72 | 56.07 | 44.56 | 54.56 | 48.81 | 57.09 | 54.04 | 59.98 | 46.97 | 56.21

Table 9: Results of joint and cross-domain adaptation results.

Model | L14 | R14 | R15 | R16
---|---|---|---|---
BERT | 48.12 | 58.23 | 60.02 | 59.91
ALBERT | 48.35 | 61.20 | 60.85 | 61.76
RoBERTa | 47.69 | 58.21 | 59.87 | 58.65
ELECTRA | 51.76 | 61.39 | 61.25 | 60.95
DeBERTa | 49.56 | 57.12 | 60.34 | 62.77
T5 | 34.54 | 39.31 | 43.69 | 44.17
Pythia | 47.83 | 57.87 | 58.21 | 60.06
GPTNeo | 45.12 | 58.33 | 60.17 | 59.25
Cerebras | 46.53 | 55.96 | 59.88 | 57.53

Table 10: Results of the AOOSPE reporting the accuracy scores.

##  Appendix D Extended Related Work

LMs and deep learning methods have been used for a plethora of downstream
tasks for a long time. Several recent works have leveraged NLP methods and
simple sampling methods for different downstream results The study of whether
existing LMs can understand instructions has motivated a range of subsequent
works. Gupta et al. (2022) showed the feasibility of LLM’s understanding of
context. This was leveraged extensively for creating the attention maps for
opinion word extraction as well as sentiment polarity assignment. Furthermore,
recent approaches have been proposed to improve the use of targeted data
generation for synthetic data creation which could be used as a cost-effective
alternative for data labelling Gupta et al. (2023).


Basic finite group theory via the lifting property

Formulating basic notions of
finite group theory
via the lifting property
masha gavrilovich]masha gavrilovich
Institute for Regional Economic Studies, Russian Academy of Sciences (IRES RAS).
National Research University Higher School of Economics, Saint-Petersburg.

<EMAIL_ADDRESS><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).

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
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.

§ INTRODUCTION.

We observe that several
elementary notions of finite group theory
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.

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.<ref>-<ref>.

In <ref> we reformulate as a labelled commutative diagram the conjecture
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>, Corollary <ref> and <ref>), 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.

Such reformulations lead one to the following questions:

* Can one extend this notation to capture more of finite group theory?

* Is this a hint towards
category theoretic point of view on finite group theory?

If one believes the evidence
provided by our examples
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.

§.§.§ Motivation
Our motivation was to formulate
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.

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?

§ DEFINITIONS AND EXAMPLES OF REFORMULATIONS

§.§ 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,

is used to define properties of morphisms starting from an explicitly given class of morphisms.

We say that two morphisms $A \xra f B$ and $X \xra g Y$ in a category $C$ are orthogonal and write $f\rtt g$
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 $\tilde j\circ g=j$ (see Figure 1a).

We may also say that $f$ lifts wrt $g$, $f$ left-lifts wrt $g$, or $g$ right-lifts wrt $f$, or that $f$ antagonizes $g$.

By analogy with orthogonal complement of a non-symmetric bilinear form,
define  left/right Quillen negation or left/right Quillen orthogonal of a class $P$
of morphisms:

$$ P^{\rtt l}:=\{\, f\ :\ f\,\rtt\,g\,\ \text{for each }g\in P\,\}$$

$$P^{\rtt r}:=\{\, g\ :\ f\, \rtt g\,\ \text{for each }g\in P\,\}$$

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 without a given property
in a manner useful in a category theoretic diagram chasing computation.

#1#2#3#4#5#6 #1 [r]^ @->[d]_#2     #4 [d]^#5
#3  [r] @–>[ur]^    #6

$(a)\  \xymatrix{ A \ar[r]^{i} \ar@{->}[d]_f & X \ar[d]^g \\ B \ar[r]_-{j} \ar@{-->}[ur]^{{\tilde j}}& Y }$
$\ \ \ \ \ \ (b)\ \xymatrix{ A \ar[r] \ar@{->}[d]_{(P)} & X \ar[d]^{\therefore (Q)} \\ B \ar[r] \ar@{-->}[ur]& Y }$
$\ \ \ \ \ \ (c)\ \xymatrix{ A \ar[r] \ar@{->}[d]_{\therefore(P)} & X \ar[d]^{ (Q)} \\ B \ar[r] \ar@{-->}[ur]& Y }$

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\,\}$

§.§ 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.

In the category of Finite Groups,

* $(\AbKer)^\lr$ is the class of homomorphisms whose kernel is solvable

* $(0\lra *)^\lr$ is the class of subnormal subgroups

* $(0\lra \Ab)^\lr=(0\lra \Ab)^\lr=\{[G,G]\lra G: G\text{ is arbitrary}\}^\lr$ is the class of subgroups $H<G$ such that there is a chain of subnormal subgroups
$H=G_0 \vartriangleleft G_1  \vartriangleleft \ldots  \vartriangleleft G_n =G$ such that $G_{i+1}/G_{i}$ is Abelian, for $i=0,...,n-1$.

* $(0 \lra S)^\lr$ is the class of subgroups $H<G$ such that there is a chain of subnormal subgroups
$H=G_0 \vartriangleleft G_1  \vartriangleleft \ldots  \vartriangleleft G_n =G$ such that $G_{i+1}/G_{i}$ embeds into $S$, for $i=0,...,n-1$.

* $(\ZZ/p\ZZ\longrightarrow 0)^{r}$ is the class of homomorphisms whose kernel has no elements of order $p$

* $(\ZZ/p\ZZ\longrightarrow 0)^{\rr}$ is the class of surjective homomorphisms whose kernel is a $p$-group

In the category of Groups,

* $(*\lra 0)^{\rtt l}$ is the class of retracts

* $(0\lra *)^{\rtt r}$ is the class of split homomorphisms

* $(0\longrightarrow \ZZ)^{r}$ is the class of surjections

* $(\ZZ\lra 0)^{r}$ is the class of injections

* a group  is free iff
$0\longrightarrow F$ is in  $(0\longrightarrow \ZZ)^{rl}$

* a group $A$ is Abelian iff $A\lra 0$ is in $( \Bbb F_2 \lra \ZZ\times\ZZ)^{\rtt r}$

* group $G$ can be obtained from $H$ by adding commutation relations, i.e. the kernel of $H\lra G$ is generated by commutators $[h_1,h_2]$, $h_1,h_2\in H$,
$H\lra G$ is in $( \Bbb F_2 \lra \ZZ\times\ZZ)^{rl}$

* subgroup $H$ of $G$ is a normal span of substitutions in words $w_1,..,w_i$ of the free group $\Bbb F_n$ iff $G \lra G/H$ is in $( \Bbb F_n \lra \Bbb F_n/<w_1,...,w_i>)^{rl}$

* $(\AbKer)^{\rtt l}$ is the class of homomorphisms whose kernel is perfect

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 $C<C'$ be a minimal subnormal subgroup of $D$ containing $C$.
Then $C\lra C'$ is in $(0\lra *)^{\rtt l}$ and the lifting property $C\lra C' \rtt C\lra D$ fails, as required.

Items 5 and 6 use Cauchy theorem that a prime $p$ divides the order of a group iff the group has an element of order $p$.

§.§ Concise reformulations in terms of iterated Quillen negation

We use the observations above to concisely reformulate several elementary notions in finite group theory.

In the category of Finite Groups,

* a finite group $S$ is soluble iff either of the following equivalent conditions holds:

* $S\lra 0$ is in $(\AbKer)^\lr$

* $0\lra S$ is in $(0\lra \Ab)^\lr$

* a finite group $H$ is nilpotent iff

* the diagonal map $H\lra H\times H$, $x\mapsto (x,x)$, is in  $(0\lra *)^\lr$

* the Fitting subgroup $F$ of $G$ is the largest subgroup such that

* the diagonal map $F\lra G\times G$, $x\mapsto (x,x)$, is in  $(0\lra *)^\lr$

* a finite group $H$ is a $p$-group iff one of the following equivalent conditions hold:

* $H\lra 0$ is in  $(\ZZ/p\ZZ\lra 0)^\rr$

* $0\lra H$ is in $(0\lra \ZZ/p\ZZ)^\lr$

In the category of  Groups,

* a group $G$ is torsion-free iff  iff $0\longrightarrow G$ is in $\{ n\ZZ\longrightarrow \ZZ: n>0 \}^{\rtt r}$

* a subgroup  contains torsion and is pure iff $H\longrightarrow G$ is in $\{ n\ZZ\longrightarrow \ZZ: n>0 \}^{\rtt r}$

* $H$ 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

$$H \lra G \xra {(\Bbb F_n \lra\, \Bbb F_n/<w_1,..,w_i>)^{\rtt rl}} \cdot \xra{(\Bbb F_n \lra\, \Bbb F_n/<w_1,..,w_i>)^{\rtt r}} 0,$$
or, equivalently, is the kernel of the corresponding homomorphism

* a finite 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 $\gamma<\beta$,

* $S\lra 0$ is in $(\AbKer)^\lr$

* 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 $\gamma<\beta$,

* the diagonal map $H\lra H\times H$, $x\mapsto (x,x)$, is in  $(0\lra *)^\lr$

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.

§.§ $p$-, $p'$-, 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 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.

[$p$-core, $p'$-core, $p,p'$-core] In the category of Finite Groups,

* 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

$$ G \xra {(\ZpZ\lra 0)^\rr} G/O_p(G) \xra{(\ZpZ\lra 0)^{\rtt rrl}} 0$$

* 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

$$ G \xra {(\ZpZ\lra 0)^{\rtt r}} G/O_{p'}(G) \xra{(\ZpZ\lra 0)^{\rtt rl}} 0$$

* the $p,p'$-core $O_{p,p'}(G)=O_p(G/O_{p'}(G))$ of $G$
is the group appearing in the unique decomposition of form

$$ G \xra {(\ZpZ\lra 0)^{\rtt r}} G/O_{p'}(G)  \xra {(\ZpZ\lra 0)^{\rtt rr}} G/O_p(G/O_{p'}(G))   \xra{(\ZpZ\lra 0)^{\rtt rl}} 0$$

We end with a couple of test questions suggested by Bob Oliver (private communication).

Can the Sylow theorems (only existence and uniqueness of Sylow subgroups)
be proven using the characterization of p-groups by Corollary <ref>(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?

§.§ $f$-local groups, localisations and nilpotent groups.

[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].]

Let $ f \uurtt g$ denote the 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 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
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 $H\lra H\times H$.

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.

Our notation suggests the following modifications of the conjecture.

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

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

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

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.

Under what assumptions on
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?

§.§ 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 Abelian iff

\[ \left<a,b\right> \,\lra\,\left<a,b:ab=ba\right>  \rtt\,\, A\lra 0
\]

where $\left<a,b\right> \,\lra\,\left<a,b:ab=ba\right>$ is the  abelianisation morphism sending the free group into the Abelian free group on two generators;
a group $G$ is perfect, $G=[G,G]$, iff $G\nto A$ for any Abelian group $A$, i.e.

\[ \left<a,b\right> \,\lra\,\left<a,b:ab=ba\right>  \rtt\,\, A\lra 0\ \implies\ G\lra 0 \rtt A\lra 0\]

equivalently, for an arbitrary homomorphism $g$,

\[ \left<a,b\right> \,\lra\,\left<a,b:ab=ba\right>  \rtt\,\,g \ \ \implies\ G\lra 0 \rtt\, \,g\,\]

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 soluble iff  $G\nto H$ for each perfect group $G$,

$$0\lra G\,\rtt 0\lra H\text{ or equivalently }G\lra 0 \rtt H\lra 0.$$

Alternatively, a group $H$ is 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

$$D=G_0 \vartriangleleft G_1  \vartriangleleft \ldots  \vartriangleleft G_n =G$$

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
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

§ SPECULATIONS. DIAGRAMS COMMUTING UP TO CONJUGATION

In this section we make some speculation and remarks on ways to extend our notation.

It is useful to consider diagrams which commute 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]:

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 $H\lra H, x\mapsto \iota(g)x\iota(g^{-1})$

* 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 $H\lra H,\ x\mapsto \iota^{-1}(g)x\iota^{-1}(g^{-1})$

$(a)\  \xymatrix{ A \ar[r]^{i} \ar@{->}[d]_f & X\ar@{-->}[r]^\sigma & X^\sigma \ar[d]|--{\sigma g\sigma^{-1}} \\ B \ar[r]|-{j} \ar@{-->}[urr]|{{\tilde j}}& Y \ar@{-->}[r]^\sigma & Y }$
$(b)\  \xymatrix{ P_1 \ar[r] \ar@{->}[d]|{(0\lra\ZpZ)^{\rttt lr}}  & S_p\ar@{-->}[r]^\sigma & S_p^\sigma \ar[d]|--{\sigma g\sigma^{-1}}
\\ P_2 \ar[r]  \ar@{-->}[urr]  & G \ar[r]^\sigma & G }$

The definition of a lifting property $f\rttt g$ up to conjugation.

(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.

Say that two morphisms $A \xra f B$ and $X \xra g Y$ in a category $C$ are orthogonal up to conjugation and write $f\rttt g$
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 $\tilde j:B\lra X$
and an inner automorphism $\sigma:Y\lra Y$ such that
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).

Define left/right Quillen negation or left/right Quillen orthogonal $P^{\rttt l}$, $P^{\rttt r}$ up to conjugation
in the obvious way.

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.

#1#2#3#4#5#6#7 #1 [r]^ @->[d]_#2     #4 [d]_#5^#6
#3  [r] @–>[ur]^    #7

$(a)\  \rrt  {0} {} {\Bbb Z}   H {\therefore(surjective)} G $

$(b)\  \rrt   {\Bbb Z} {} 0  H {\therefore(injective)} G $
$(c)\  \rrt  { \left<a,b\right>} {.} {\left<a,b:ab=ba\right>}   A {\therefore(abelian)} 0 $

$(d)\  \rrt {G} {\therefore(perfect)} {0}  A {(abelian)} 0 $
$(e)\  \rrt G {(perfect)} {0} {H} {\therefore(soluble)} {0}$
$(f)\  \rrt {{\Bbb Z}/p{\Bbb Z}} {.} {0} {H} {\therefore(\,p\,\nmid\,\textrm{card}H)} {0}$

$(g)\ \rrt {H} {(\,p\,\nmid\,\textrm{card}H)} {0} {G} {\therefore(\textrm{card}H=p^n\text{ for some }n)} {0}$

$(h)\  \rrt {{\Bbb Z}/2{\Bbb Z}} {.} {0} {H} {\therefore(odd)} {0}$
$(i)\ \rrt {{\Bbb Z}/2{\Bbb Z}} {.} {0} {H} {\therefore(soluble)} {0}$
$(k)\ \rrt  {N} {\therefore(normal\ closure)} {H} {0} { } {G}$
$(l)\ \rrtt   {N} {(normal\ closure)} {H} {D} {} { \therefore(subnormal)  } {G}$
$(m)\ \rrtt   {N} {(normal\ closure)} {H} {G} {\Delta} { \therefore(nilpotent)  } {G\times G}$

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.(a) a homomorphism $H\lra G$ is surjective, i.e. for each $g\in G$ there is $h\in H$ sent to $g$(b) a homomorphism $H\lra G$ is injective, i.e. the kernel of $H\lra G$ is the trivial group(c) a group is abelian iff each morphism from the free group of two generators
factors through its abelianisation ${\Bbb Z}\times {\Bbb Z}$.

(d)  a group $G$ is perfect, $G=[G,G]$, iff it admits no non-trivial homomorphism to an abelian group(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.
(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 $p$(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 $p$(h)  by Cauchy's theorem, a finite group has an odd number of elements iff it contains no involution $e,e^2=1, e\neq1$(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.(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 $G$(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 $N$(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 $N\lra H$

§ ACKNOWLEDGMENTS AND 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.

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 …

[1]
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Self-derived localizations of groups.

[4]
C. Casacuberta, D. Scevenels, and J. H. Smith.
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<http://groupprops.subwiki.org/wiki/Nilpotent group>