00footnotetext: 2010 AMS Subject Classification: Primary 20K10, 20K12. Key
words and phrases: Abelian groups, fully invariant subgroups, characteristic
subgroups.

# On Abelian Groups Having All Proper
Characteristic Subgroups Isomorphic

Andrey R. Chekhlov
Faculty of Mathematics and Mechanics, Section of Algebra,
Tomsk State University, Tomsk 634050, Russia
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
and
Peter V. Danchev
Institute of Mathematics and Informatics, Section of Algebra,
Bulgarian Academy of Sciences, Sofia 1113, Bulgaria
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider two variants of those Abelian groups with all proper
characteristic subgroups isomorphic and give an in-depth study of their basic
and specific properties in either parallel or contrast to the Abelian groups
with all proper fully invariant subgroups isomorphic, which are studied in
details by the current authors in Commun. Algebra (2015). In addition, we also
examine those Abelian groups having at least one proper characteristic
subgroup isomorphic to the whole group. The established by us results somewhat
extend those obtained by Grinshpon-Nikolskaya in Tomsk State Univ. J. Math. &
Mech. (2011, 2012) and in Commun. Algebra (2011), respectively.

##  1 Introduction and Definitions

Throughout this article, let all groups into consideration be additively
written and abelian. Our notations and terminology from group theory are
mainly standard and follow those from [3], [4] and [8], respectively. Another
useful source on the explored subject is [10] as well. For instance, if $p$ is
a prime integer and $G$ is an arbitrary group, $p^{n}G=\\{p^{n}g~{}|~{}g\in
G\\}$ denotes the $p^{n}$-th power subgroup of $G$ consisting of all elements
of $p$-height greater than or equal to $n\in\mathbb{N}$, $G[p^{n}]=\\{g\in
G~{}|~{}p^{n}g=0,n\in\mathbb{N}\\}$ denotes the $p^{n}$-socle of $G$, and
$G_{p}=\cup_{n<\omega}G[p^{n}]$ denotes the $p$-component of the torsion part
$tG=\oplus_{p}G_{p}$ of $G$.

On the other hand, if $G$ is a torsion-free group and $a\in G$, then let
$\chi_{G}(a)$ denote the characteristic and let $\tau_{G}(a)$ denote the type
of $a$, respectively. Specifically, the class of equivalence in the set of all
characteristics is just called type and we write $\tau$. If
$\chi_{G}(a)\in\tau$, then we write $\tau_{G}(a)=\tau$, and so
$\tau(G)=\\{\tau_{G}(a)~{}|~{}0\neq a\in G\\}$ is the set of types of all non-
zero elements of $G$. The set $G(\tau)=\\{g\in
G~{}|~{}\tau(g)\geqslant\tau\\}$ forms a pure fully invariant subgroup of the
torsion-free group $G$. Recall that a torsion-free group $G$ is called
homogeneous if all its non-zero elements have the same type.

Concerning ring theory, suppose that all rings which we consider are
associative with identity element. For any ring $R$, the letter $R^{+}$ will
denote its additive group. To simplify the notation and to avoid a risk of
confusion, we shall write ${\rm E}(G)$ for the endomorphism ring of $G$ and
$\mathop{\rm End}\nolimits(G)={\rm E}(G)^{+}$ for the endomorphism group of
$G$.

As usual, a subgroup $F$ of a group $G$ is called fully invariant if
$\phi(F)\subseteq F$ for any $\phi\in{\rm E}(G)$, while if $\phi$ is an
invertible endomorphism (= an automorphism), then $F$ is called a
characteristic subgroup.

Classical examples of important fully invariant subgroups of an arbitrary
group $G$ are the defined above subgroups $p^{n}G$ and $G[p^{n}]$ for any
natural $n$ as well as $tG$ and the maximal divisible subgroup $dG$ of $G$;
actually $dG$ is a fully invariant direct summand of $G$ (see, for instance,
[3]).

To avoid any confusion and misunderstanding, we shall say that a group $G$ has
only trivial fully invariant subgroups if $\\{0\\}$ and $G$ are the only ones.
Same appears for the characteristic subgroups, respectively.

The following notions were stated in [2].

Definition 1. A non-zero group $G$ is said to be an IFI-group if either it has
only trivial fully invariant subgroups, or all its non-trivial fully invariant
subgroups are isomorphic otherwise.

Definition 2. A non-zero group $G$ is said to be an IC-group if either it has
only trivial characteristic subgroups, or all its non-trivial characteristic
subgroups are isomorphic otherwise.

Note that Definition 2 implies Definition 1. In other words, any IC-group is
an IFI-group; in fact every fully invariant subgroup is characteristic.

Definition 3. A non-zero group $G$ is called a strongly IFI-group if either it
has only trivial fully invariant subgroups, or all its non-zero fully
invariant subgroups are isomorphic otherwise.

Definition 4. A non-zero group $G$ is called a strongly IC-group if either it
has only trivial characteristic subgroups, or all its non-zero characteristic
subgroups are isomorphic otherwise.

Notice that Definition 4 implies Definition 3.

On another vein, Definition 4 obviously yields Definition 2, but the converse
is false. In fact, in [8] was constructed a single non-trivial characteristic
subgroup of a $2$-group that are pairwise non-isomorphic, thus giving an
example of an IC-group which is surely not a strongly IC-group; in other
words, Definition 4 properly implies Definition 2.

Our goal in the present paper is to explore some fundamental and exotic
properties of the defined above classes of groups, especially the IC-groups
and the strongly IC-groups. In addition, we shall investigate even something
more, namely the existence of a non-trivial characteristic subgroup of a given
group which subgroup is isomorphic to the whole group, calling these groups
weakly IC-groups. Our motivation to do that is to exhibit and compare the
characteristically similarities and discrepancies of these group classes.

The major results established by us are formulated and proved in the next
section.

##  2 Main Theorems and Examples

For completeness of the exposition and for the reader’s convenience, we first
and foremost will give a brief retrospection of the most principal results
achieved in [2] concerning IFI-groups and strongly IFI-groups.

As usual, the symbol $\oplus_{m}G=G^{({m})}$ will denote the external direct
sum of $m$ copies of the group $G$, where $m$ is some ordinal (finite or
infinite).

###### Theorem 2.1

Let $G$ be a $p$-group and let $m\geqslant 2$ be an ordinal. Then $G^{({m})}$
is an IFI-group if and only if $G$ is an IC-group if and only if $G$ is an
IPI-group.

###### Proposition 2.2

Let $G$ be a torsion-free group. Then $G$ is an IFI-group if and only if $G$
is a strongly IFI-group.

###### Lemma 2.3

(a) A fully invariant subgroup of an IFI-group is an IFI-group.

(b) A fully invariant subgroup of a strongly IFI-group is a strongly IFI-
group.

###### Proposition 2.4

A non-zero IFI-group is either divisible or reduced.

###### Theorem 2.5

The following two points hold:

(i) A non-zero group $G$ is an IFI-group if and only if one of the following
holds:

$\bullet$ For some prime $p$ either $pG=\\{0\\}$, or $p^{2}G=\\{0\\}$ with
$r(G)=r(pG)$.

$\bullet$ $G$ is a homogeneous torsion-free IFI-group of an idempotent type.

(ii) A non-zero torsion group $G$ is a strongly IFI-group if and only if it is
an elementary $p$-group for some prime $p$.

###### Proposition 2.6

Every homogeneous fully transitive torsion-free group of an idempotent type is
an IFI-group.

###### Corollary 2.7

A direct summand of a fully transitive torsion-free IFI-group is again a fully
transitive IFI-group.

We next continue with the statements and proofs of the chief results. Before
doing that, we record the following two hopefully useful assertion.

###### Lemma 2.8

If $G$ is a decomposable fully transitive homogeneous torsion-free group, then
all its characteristic subgroups are fully invariant.

Proof. Let us decompose $G=A\oplus B$, where $A,B\neq\\{0\\}$, $H\leq G$ is a
characteristic subgroup and $\pi:G\rightarrow A$ is the corresponding
projection. If $x\in\pi(H)$, $f\in\mathrm{E}(A)$ and $y=f(x)$, then there
exists an element $z\in B$ with $\chi(z)=\chi(y)$. So, $\varphi(y)=z$,
$\psi(z)=y$ for some $\varphi\in\mathrm{Hom}(A,B)$,
$\psi\in\mathrm{Hom}(B,A)$. Applying Lemma 2.15 appointed below, we get that
$y\in H$. Employing once again Lemma 2.15 (1), we can deduce the wanted full
invariability of $H$.

###### Lemma 2.9

If $G$ is a homogeneous torsion-free transitive group, then all its
characteristic subgroups are fully invariant.

Proof. Choose $0\neq x\in H$ and $f\in\mathrm{E}(G)$, where $H$ is
characteristic in $G$. We have $\chi(x)\leq\chi(f(x))$ and $t(x)=t(f(x))$
whenever $f(x)\neq 0$, so that $\chi(kx)=\chi(f(x))$ for some natural number
$k$, whence $\varphi(kx)=f(x)$, where $\varphi\in\mathrm{Aut}(G)$, and so
$f(x)\in H$, as needed.

###  2.1 IC-groups

We start our work here with some elementary but useful observations:

$\bullet$ Since the divisible part of any group is necessarily fully
invariant, a non-zero IC-group is either divisible or reduced, and hence it
follows from Theorem 2.5 that it is either a $p$-group or a homogeneous
torsion-free group of an idempotent type.

$\bullet$ If $G$ is a torsion-free group, then $G$ is an IC-group if, and only
if, $G$ is a strongly IC-group (compare with the next subsection). The proof
is as Proposition 2.2 from [2].

$\bullet$ Every homogeneous transitive torsion-free group of an idempotent
type is an IC-group – in fact, in view of the condition being homogeneous, we
know that such a group is necessarily fully transitive.

$\bullet$ Any divisible group is an IC-group if, and only if, it is a torsion-
free group. (This follows automatically from [2, Proposition 2.7].)

We now continue with the following technicalities.

###### Lemma 2.10

(1) If $p^{2}G=\\{0\\}$ for some prime $p$, then every characteristic subgroup
$H$ in $G$ is fully invariant.

(2) Every torsion IFI-group is an IC-group.

Proof. (1) We have $G=G_{1}\oplus G_{2}$, where $G_{1}$ is elementary and
$G_{2}\cong\bigoplus\mathbb{Z}(p^{2})$ whenever $pG\neq\\{0\\}$. If
$pH\neq\\{0\\}$, then as in Proposition 2.20 $H=G$. So, let us assume that
$pH=\\{0\\}$. Supposing $\pi:G\rightarrow G_{1}$ is the corresponding
projection and $\pi(H)\neq\\{0\\}$, it follows directly from Lemma 2.15 that
$H=G[p]$. However, if $\pi(H)=\\{0\\}$, then applying once again the same
Lemma 2.15, we extract that $H=G_{2}[p]=pG[p]$. So, in both cases, the
subgroup $H$ is fully transitive, as asserted.

(2) follows from (1).

It is an evident fact that if any element of the endomorphism ring
$\mathrm{E}(G)$ is an integer multiplied by invertible, then all fully
invariant subgroups of a given group are always characteristic.

Furthermore, by usage of [2, Theorem 2.12], if $G$ is a torsion-free IFI-group
whose non-zero endomorphisms are monomorphisms, then $G$ is an E-group, so if
the rank of the additive group $\mathrm{E}(G)^{+}$ is finite for such a group
$G$, then the tensor product $\mathrm{E}(G)\otimes_{\mathbb{Q}}\mathbb{Q}$ is
a field, and every endomorphism of $G$ is an integer multiplied by invertible
(see the paragraph before Theorem 4.7 in [9]). Thus, one can state the
following direct consequence:

###### Corollary 2.11

If $G$ is a torsion-free IFI-group of finite rank whose non-zero endomorphisms
are monomorphisms, then $G$ is an IC-group.

However, in the case of torsion-free groups of infinite rank the situation is
totally different as the next construction shows. Before listing it, we need
the following technicalities.

If the additive group $R^{+}$ of a ring $R$ is a torsion-free group, then we
shall say that the ring $R$ is a torsion-free ring. Likewise, we put
$\pi(R)=\\{p\in\mathbb{P}\,|\,pR\neq R\\}$, where $h_{p}^{G}(x)$ is the
$p$-height of an element $x$ in $G$.

###### Lemma 2.12

[9, Lemma 44.6] The following properties of a torsion-free ring $R$ are
equivalent:

(1) For every $p\in\pi(R)$ and all $a,b\in R$, if $p\,|\,ab$, then (either)
$p\,|\,a$ or $p\,|\,b$;

(2) The equality $h_{p}(ab)=h_{p}(a)+h_{p}(b)$ holds for each $p\in\pi(R)$ and
all $a,b\in R$;

(3) The ring $R/pR$ has no zero divisors.

We are now ready to give our example.

###### Example 2.13

There exists an IFI-group which is not an IC-group.

Proof. Let $\sigma\in\hat{\mathbb{Z}}_{p}$ is invertible, but transcendental
on $\mathbb{Z}$, and $R$ is a pure subring in $\hat{\mathbb{Z}}_{p}$ generated
by $1$ and $\sigma$; standardly, $\hat{\mathbb{Z}}_{p}$ is the ring of
$p$-adic integers. If we set $G=R^{+}$, then it is obviously an $E$-group. We
now intend to illustrate that $R$ is a ring of principal ideals, whence $G$
will be an IFI-group by consulting with [2, Theorem 2.12]. To that objective,
given $I\lhd R$ is an arbitrary ideal. If $\alpha\in R$, then
$p^{n}\alpha\in\mathbb{Q}_{p}[\sigma]$, where as usual $\mathbb{Q}_{p}$
designates the ring of all rational numbers whose denominators are not divided
by $p$, and $\mathbb{Q}_{p}[\sigma]$ is the ring of polynomials of $\sigma$
with coefficients from $\mathbb{Q}_{p}$. Letting $\gamma$ be the polynomial
”of least degree” coming $I$, then one sees that $\gamma R\leq I$. Notice also
that $I$ is a $q$-pure ring (without identity) in $R$ for every prime number
$q\neq p$. Thus, for every $\beta\in I$, it follows that $p^{m}\beta\in\gamma
R$ for some $m\in\mathbb{N}$. Indeed, if
$\beta=b_{0}\sigma^{k}+b_{1}\sigma^{k-1}+\dots+b_{k}$,
$\gamma=c_{0}\sigma^{s}+c_{1}\sigma^{s-1}+\dots+c_{s}$, where
$b_{i},c_{j}\in\mathbb{Q}_{p}$, $k,s\in\mathbb{N}$, $c_{0}=p^{t}d_{0}$, where
$d_{0}\in\mathbb{Q}_{p}\setminus p\mathbb{Q}_{p}$, then $k\geq s$ and
$p^{t}\beta-(b_{0}d_{0}^{-1}\sigma^{k-s})\gamma$ has degree on $\sigma$
smaller than $k$, so that we can use an induction. Suppose
$\gamma_{0}=p^{s}\gamma$, where $\gamma_{0}\in\hat{\mathbb{Z}}_{p}\setminus
p\hat{\mathbb{Z}}_{p}$. Hence $\gamma_{0}R$ is a pure subring in $R$. In fact,
as $R$ is pure in $\hat{\mathbb{Z}}_{p}$, we have that
$R/pR\cong\hat{\mathbb{Z}}_{p}/p\hat{\mathbb{Z}}_{p}$, an application of Lemma
2.12 gives that
$h_{p}^{R}(\gamma_{0}r)=h_{p}^{R}(\gamma_{0})+h_{p}^{R}(r)=h_{p}^{R}(r)$,
because $h_{p}^{R}(\gamma_{0})=0$.

Furthermore, it follows from the noted above properties of elements of $I$
that $I\leq\gamma_{0}R$. But $\gamma_{0}R/\gamma R\cong\mathbb{Z}_{p^{s}}$ and
since $I/\gamma R\leq\gamma_{0}R/\gamma R$, one verifies that $I/\gamma
R\cong\mathbb{Z}_{p^{l}}$ for some $l\leq s$. In particular,
$I=(p^{l}\gamma)R$, i.e., $I$ is a proper principal ideal of $R$, as expected,
and hence $G$ is an $IFI$-group, as indicated.

Moreover, note that $\mathrm{Aut}(G)$ is isomorphic to the group
$U(\mathbb{Q}_{p})$ consisting of all invertible elements of the ring
$\mathbb{Q}_{p}$. To show that, assume $\alpha\in R$ is invertible, and since
$p^{n}\alpha\in\mathbb{Q}_{p}[\sigma]$ it must be that
$p^{n}\alpha=a_{0}\sigma^{k}+a_{1}\sigma^{k-1}+\dots+a_{k}$ for some
$k\in\mathbb{N}$ and $a_{0},a_{1},\dots,a_{k}\in\mathbb{Q}_{p}$. Similarly,
$p^{m}\alpha^{-1}=b_{0}\sigma^{s}+b_{1}\sigma^{s-1}+\dots+b_{s}$. Therefore,
$(a_{0}\sigma^{k}+a_{1}\sigma^{k-1}+\dots+a_{k})(b_{0}\sigma^{s}+b_{1}\sigma^{s-1}+\dots+b_{s})=p^{n+m}$,
which is actually the desired contradiction that $\sigma$ is transcendental
provided $k\neq 0$. Consequently, in $G$ there will exist a characteristic
subgroup of rank $1$, but since $G$ is supposed to be infinite rank it
definitely will not be an IC-group, as claimed.

In view of [2, Corrolary 2.22], one can say that if $G$ is an IFI-group, then
the cartesian direct sum $G^{(m)}$ is also an IFI-group for any ordinal $m$.
However, the following is true:

###### Example 2.14

There exists such a group $A$ that $G=A^{(m)}$ is an $IC$-group for every
ordinal $m>1$, but $A$ is not an $IC$-group. In particular, the direct summand
of an $IC$-group need not be an $IC$-group.

Proof. Take the group $A$ from Example 2.13. Thus, $G$ is an $IFI$-group,
because $A$ is an $IFI$-group. Now, suppose that $H\leq G$ is a characteristic
subgroup. Since every endomorphism of $G$ is a sum of automorphisms, it must
be that $H$ is fully invariant. Consequently, $H\cong G$, i.e., $G$ is an
$IC$-group, as asserted.

###  2.2 Strongly IC-groups

We start our work here with some elementary but helpful observations:

$\bullet$ A non-zero torsion group $G$ is a strongly IC-group if, and only if,
it is an elementary $p$-group for some prime $p$. (This follows immediately
from Theorem 2.5 quoted above.)

In virtue of Theorem 2.5, one infers that a strongly IFI-group is exactly a
strongly IC-group, as well as a torsion-free strongly IC-group is precisely a
torsion free IC-group.

###  2.3 Weakly IC-groups

Before starting our work, it is worthwhile noticing that in [5], [6] and [7]
were investigated those Abelian groups (both torsion and torsion-free) having
isomorphic proper fully invariant subgroup. In this subsection, we will
initiate an examination of the case when we have a proper characteristic
subgroup isomorphic to the whole group. For simplicity of the exposition, we
shall call these groups just weakly IC-groups.

We now continue with our principal results. Specifically, we proceed by
proving with the following statements.

###### Lemma 2.15

Let $G=A\oplus B$ and $\pi:G\rightarrow A$ be the corresponding projection. If
$H$ is a characteristic subgroup in $G$, then:

(1) The inclusion $f\pi(H)\leq H$ holds for every $f\in\mathrm{Hom}\,(A,B)$.

(2) $\pi(H)$ is a characteristic subgroup in $A$.

Proof. (1) It is obvious that the endomorphism
$\left(\begin{array}[]{cc}1&0\\\ f&1\end{array}\right)$ is an automorphism of
$G$, hence, for any $x=a+b\in H$ with $a\in A$, $b\in B$, we have
$f(x)=x+f(a)$. Thus, $f(a)=f\pi(x)\in H$, as required.

(2) If $\alpha\in\mathrm{Aut}(A)$, then one checks that
$\left(\begin{array}[]{cc}\alpha&0\\\
0&1\end{array}\right)\in\mathrm{Aut}(G)$, hence, for any $a+b\in H$ with $a\in
A$, $b\in B$, we have $\alpha(a)+b\in H$. So, $a-\alpha(a)\in H$ and,
therefore, $\alpha(a)=a-(a-\alpha(a))\in\pi(H)$, as required.

###### Proposition 2.16

(1) Let $G=\bigoplus_{i\in I}G_{i}$, where, for each direct summand $G_{i}$,
there is an other summand $G_{j}$ with $j\in I\setminus\\{i\\}$ such that
$G_{i}\cong G_{j}$. Then every characteristic subgroup of $G$ is fully
invariant.

(2) Let $A=B\oplus G$, where $G$ is such a decomposable group that, for every
its direct summand $C$, it decomposes as $G=C\oplus K$ and in the complement
$K$ there exists a direct summand isomorphic to $C$. Then, if
$\pi:A\rightarrow G$ is projection and $H$ is a characteristic subgroup in
$A$, it follows that $\pi(H)=H\cap G$ and $H\cap G$ is fully invariant in $G$.

Proof. (1) In view of the isomorphism between $G_{i}$ and $G_{j}$ and
consulting with Lemma 2.15, we deduce that not only that $\pi_{i}(H)\leq H$
for the projection $\pi_{i}:G\rightarrow G_{i}$, but also that
$\varphi\pi_{i}(H)\leq H$ for every endomorphism $\varphi$ of $G_{i}$ and,
besides, that $f\pi_{i}(H)\leq H$ for every
$f\in\mathrm{Hom}\,\bigl{(}G_{i},\bigoplus_{j\neq i}G_{j}\bigr{)}$. This
surely means that the subgroup $H$ is fully invariant, as claimed.

(2) follows directly from (1).

We can now say slightly more in the case of arbitrary direct sums.

###### Lemma 2.17

Let $G=\bigoplus_{i\in I}G_{i}$ and $\pi_{i}:G\rightarrow G_{i}$ be the
corresponding projections, where $I$ is an arbitrary index set. If $H$ is a
characteristic subgroup in $G$, then both the subgroups
$\underline{H}=\bigoplus_{i\in I}(H\cap G_{i})$ and
$\overline{H}=\bigoplus_{i\in I}\pi_{i}(H)$ also are characteristic subgroups
in $G$.

Proof. Let $x\in H\cap G_{i}$ and $\alpha\in\mathrm{Aut}(G)$. Since
$\alpha(x)$ can be embedded in a sum of finite numbers of direct components
$G_{i}$, we may assume that the set $I$ is finite, thus writing
$G=G_{1}\oplus\dots\oplus G_{n}$ for some $n\geq 2$ and $x\in G_{1}$. Then,
mimicking [3],[4], we may identify $\alpha$ as the $n\times n$ matrix

$\alpha=\left(\begin{array}[]{cccc}\alpha_{11}&\alpha_{12}&\dots&\alpha_{1n}\\\
\alpha_{21}&\alpha_{22}&\dots&\alpha_{2n}\\\ \dots&\dots&\dots&\dots\\\
\alpha_{n1}&\alpha_{n2}&\dots&\alpha_{nn}\end{array}\right)$

and so we write
$\alpha(x)=\alpha_{11}(x)+(\alpha_{21}(x)+\dots+\alpha_{n1}(x))\in H$. We have
$\alpha_{11}(x)\in G_{1}$ and hence Lemma 2.15 (1) yields $\alpha_{i1}(x)\in
H\cap G_{i}$, where $i=2,\dots,n$, so that $\alpha_{11}(x)\in H\cap G_{1}$.
Consequently, $\underline{H}$ is characteristic in $G$, as claimed.

If now $x_{1}\in\pi_{1}(H)$, then $x_{1}+\dots+x_{n}=h$ for some $h\in H$ and
$x_{i}\in G_{i}$ ($i=2,\dots,n$). Therefore,

$\alpha(h)=\alpha_{11}(x_{1})+\alpha_{22}(x_{2})+\dots+\alpha_{nn}(x_{n})+\sum_{i\neq
j}\alpha_{ij}(x_{j}),$

where $\alpha(h)\in H$ and again applying Lemma 2.15 (1) it must be that
$\alpha_{ij}(x_{j})\in H\cap G_{i}$ whenever $i\neq j$. So, we get

$\pi_{1}\alpha(h)=\alpha_{11}(x_{1})+\alpha_{12}(x_{2})+\dots+\alpha_{1n}(x_{n})\in\pi_{1}(H).$

Since $\alpha_{12}(x_{2})+\dots+\alpha_{1n}(x_{n})\in\pi_{1}(H)$, one has that
$\alpha_{11}(x_{1})=\pi_{1}\alpha(x_{1})\in\pi_{1}(H)$ and
$\pi_{i}\alpha(x_{1})\in H\cap G_{i}\leq\pi_{i}(H)$, provided $i=2,\dots,n$,
i.e., $\alpha(x_{1})\in\pi_{1}(H)\oplus\dots\oplus\pi_{n}(H)$. Thus,
$\pi_{1}(H)\oplus\dots\oplus\pi_{n}(H)$ is characteristic in $G$, as asserted.

###### Remark 2.18

Note that if $G=A\oplus B$ and $\\{0\\}\neq H\leq G$ is a characteristic
subgroup with $H\nleq B$, then $H\cap A\neq\\{0\\}$ presuming the condition
$A[2]=\\{0\\}$. Indeed, there exists $h\in H$ with $h=a+b$, where $a\neq 0$.
Since $h^{\prime}=a-b$ lies also in $H$, we have $0\neq h+h^{\prime}=2a\in
H\cap A$.

However, it is constructed in Proposition 2.28, listed below, a $2$-group $G$
having a direct summand $\langle a\rangle\neq\\{0\\}$ and a characteristic
subgroup $\\{0\\}\neq H\leq G$ with $H\cap\langle a\rangle=\\{0\\}$, as
needed.

Notice that, in view of Lemma 2.17, if $G$ is a direct sum of cyclic groups,
writing $G=\bigoplus_{n\geq 1}G_{n}$, where $G_{n}$ is a direct sum of cyclic
groups of order $p^{n}$ and $\pi_{n}:G\rightarrow G_{n}$ are the corresponding
projections, and $\overline{H}=\bigoplus_{n\geq 1}\pi_{n}(H)$ for some
characteristic subgroup $H$ of $G$, then $\overline{H}$ is a fully invariant
subgroup such that $\overline{H}\leq F$ for every fully invariant subgroup $F$
with $H\leq F$. In fact, exploiting Proposition 2.28 quoted below, each
subgroup $\pi_{n}(H)$ is fully invariant in $G_{n}$ and thus it follows from
Lemma 2.15 that $\overline{H}$ is really a fully invariant subgroup in $G$.
Furthermore, since $\pi_{n}(H)\leq F$ for every fully invariant subgroup $F$
with $H\leq F$, it can be deduced that $F$ contains $\overline{H}$, as
expected.

Note also that each fully invariant subgroup $H$ of $G$ is closed in $G$, that
is, the quotient-group $G/H$ is separable. Indeed, this fact follows from the
observation that $H=\bigoplus_{n\geq 1}(H\cap G_{n})$, where the factor-groups
$G_{n}/(H\cap G_{n})$ are also a direct sum of cyclic groups. However, if we
assume that $H$ is only characteristic in $G$, this property may fail as the
next example manifestly shows (in it $G$ must be a $2$-group).

###### Example 2.19

Let $G=\langle a_{1}\rangle\oplus\langle a_{2}\rangle\oplus\dots$, where
$o(a_{i})=2^{2i-1}$, $h_{i}=a_{1}+2^{i}a_{i+1}$, $H=\langle h_{i}\,|\,i\geq
1\rangle$. Then $H$ is not a closed characteristic subgroup in $G$.

Proof. Write $a_{1}=h_{i}-2^{i}a_{i+1}$ for every $i\geq 1$, i.e.,
$\overline{0}\neq a_{1}+H\in(G/H)^{1}$, which means that $H$ is not closed in
$G$. Moreover, the subgroup $\langle h_{i}\rangle$ is characteristic in
$\langle a_{1}\rangle\oplus\langle a_{i+1}\rangle$ as it is generated by all
its elements of the kind $o(x)=2^{i+1}$, $h(x)=0$ and $h(2x)=i+1$. And since
one easily checks that the two inequalities

$f(\langle 2^{i+1}a_{i+2}\rangle)\leq 2^{i+1}a_{i+1}\leq\langle
h_{i}\rangle,\varphi(2^{i}a_{i+1})\leq\langle 2^{i+2}a_{i+2}\rangle\leq\langle
h_{i+1}\rangle$

hold for each element $f\in\mathrm{Hom}\bigl{(}\bigoplus_{j\geq i+2}\langle
a_{j}\rangle,\bigoplus_{1\leq j\leq i+1}\langle a_{j}\rangle\bigr{)}$ and each
element $\varphi\in\mathrm{Hom}\bigl{(}\bigoplus_{1\leq j\leq i+1}\langle
a_{j}\rangle,\bigoplus_{j\geq i+2}\langle a_{j}\rangle\bigr{)}$, we then can
infer that $H$ is characteristic in $G$, as promised.

The next series of three propositions are quite useful in the sequel.

###### Proposition 2.20

Every bounded $p$-group, divisible $p$-group and divisible torsion-free group
is not a weakly IC-group.

Proof. Let $G$ be a bounded $p$-group, say $p^{k+1}G=0$ with $p^{k}G\neq 0$,
and assume the contrary that $H\cong G$ is characteristic in $G$. If
$o(g)=p^{k}$ for some $g\in H$, then we have $G=\langle g\rangle\oplus A$ for
some $A\leq G$. Since $\mathrm{Hom}\,(\langle g\rangle,A)\langle g\rangle=A$,
with Lemma 2.15 at hand we derive that $H=G$, a contradiction. Furthermore, as
any divisible subgroup is a direct summand of a given group, the two remained
cases can be proved similarly.

###### Remark 2.21

It follows from Proposition 2.20 that all bounded groups and divisible groups
are not weakly IC-groups. Moreover, it is easy to see that a non-reduced group
is a weakly IC-group if, and only if, so is its reduced part. And so, we
hereafter will consider only reduced groups.

On the same vein, since each torsion-free weakly IC-group $G$ must be non-
divisible, one sees that $nG\cong G$ but $nG\neq G$ for some natural number,
so for the torsion-free weakly IC-group $G$ we shall assume that it contains
own characteristic subgroups different from $nG$ which are isomorphic to $G$
(notice that this observation held too in [7] for the case of IF-groups).

###### Proposition 2.22

An arbitrary non-reduced torsion-free group is a weakly IC-group if, and only
if, its reduced part is a weakly IC-group.

Proof. ”Necessity.” Write $G=R\oplus D$, where $R$ is the reduced part and $D$
is the divisible part of the group $G$, and suppose $\pi:G\rightarrow R$ is
the corresponding projection. Letting $H$ be a characteristic subgroup in $G$,
if $\pi(H)=\\{0\\}$, then $H\leq D$ which implies that $H=D$. So, if $H\cong
G$, then $\pi(H)\neq\\{0\\}$. However, it follows from Lemma 2.15 that $D\leq
H$. Thus, $H=(H\cap R)\oplus D$. Therefore, if $H\cong G$, then $H\cap R\cong
R$, as required.

The ”Sufficiency” is obvious.

Note that, according to [7, Corrolary 6], any fully invariant subgroup of a
homogeneous fully transitive torsion-free group $G$, isomorphic to the whole
group $G$, is of the form of $nG$ for some natural number $n$. As homogeneous
transitive torsion-free groups are always fully transitive, we obtain:

###### Corollary 2.23

Homogeneous fully transitive torsion-free groups are not weakly IC-groups.

We, therefore, obtain the following construction: chosen a homogeneous
separable torsion-free group of an arbitrary infinite power $\mathfrak{m}$, we
thus can get the existence of non-weakly IC-groups.

And since in each torsion-free group $G$ whose ring of endomorphisms is
isomorphic to $\mathbb{Z}$, any endomorphic image is a subgroup $nG$ for some
natural number $n$, we extract:

###### Proposition 2.24

The torsion-free group whose endomorphism ring is isomorphic $\mathbb{Z}$ is
not a weakly IC-group.

Note that, for any natural number $n$ and any infinite cardinal number
$\mathfrak{m}$, which is the smaller first powerfully unattainable number,
there exist such torsion-free groups of rank $n$ and torsion-free groups of
power $\mathfrak{m}$ (see [3, § 88, Excercise 8; § 89, Excercise 1]).

###### Example 2.25

There exist indecomposable torsion-free groups which are a weakly IC-group.

Proof. We shall exhibit two constructions as follows:

(1) Take the group $G$ with $\mathrm{E}(G)=\mathbb{Z}+\sqrt{-5}\mathbb{Z}$.
Then, in this group, all endomorphisms are injective and all subgroups are
characteristic. So, $G\cong\sqrt{-5}G$. Furthermore, assume that
$\sqrt{-5}G=kG$ for some natural number $k$. Consequently, $5G=k^{2}G$. Since
the numbers $5$ and $k$ are mutual simple, one checks that $kG=G$ and hence
$k=1$. However, the inequality $\sqrt{-5}G\neq G$ is true, that is the desired
contradiction. Note that such a group $G$ is possible to be chosen as given
(e.g., of rank $4$, it is exhibited in [3, § 110, Excercise 8]).

(2) Let $\hat{\mathbb{Z}}=\prod_{p\in\mathbb{P}}\hat{\mathbb{Z}}_{p}$, where
$\mathbb{P}$ is the set of all prime numbers, and $\hat{\mathbb{Z}}_{p}$ is
the ring of $p$-adic integers. In addition,
$\pi_{p}:\hat{\mathbb{Z}}\rightarrow\hat{\mathbb{Z}}_{p}$ are the
corresponding projections, $1=(\dots,1_{p},\dots)$ is the unit of
$\hat{\mathbb{Z}}$, and $\xi\in\hat{\mathbb{Z}}$ is such that
$\pi_{p}(\xi)=p\xi_{p}\in\hat{\mathbb{Z}}_{p}$, where $\xi_{p}$ is some
transcendental element on $\mathbb{Z}$. Further, we define $S=\mathbb{Z}[\xi]$
to be the subring in $\hat{\mathbb{Z}}$ induced by $\\{1,\xi\\}$, and
$R=S^{+}_{*}$ is a pure hull of $S^{+}$ in $\hat{\mathbb{Z}}$. Thus $R$ will
also be a subring in $\hat{\mathbb{Z}}$. It is readily checked that each
element of $R$ is an integer multiple of some
$a_{0}\xi^{n}+a_{1}\xi^{n-1}+\dots+a_{n}$ with $a_{i}\in\mathbb{Z}$, so the
only invertible elements in $R$ are $\pm 1$. Since $G=R^{+}$ is an $E$-group
as being a pure subgroup of the $E$-group $\hat{\mathbb{Z}}$, one derives that
any subgroup in $G$ is really characteristic. But we have $G\cong\xi^{2}G$,
and because the type of each element from $\xi^{2}G$ is surely strictly more
than the type of $\xi$, we conclude that $\xi^{2}G\neq kG$ for all
$k\in\mathbb{N}$, as required.

Note that the rings for which each element is an integer multiple of
invertible are identified as strongly homogeneous rings.

If $G$ is an arbitrary weakly IC-group, then a direct summand of $G$ cannot be
a weakly IC-group. Indeed, such a possibility occurs by taking a bounded
direct summand of $G$ (see, for instance, Proposition 2.20).

We shall now deal with primary groups by starting with the following example
which manifestly illustrates that the class of weakly IC-groups is not closed
under the formation of direct summands.

###### Example 2.26

There exists such a $p$-primary weakly IC-group $G$ that some unbounded direct
summand of $G$ is not a weakly IC-group.

Proof. By virtue of [6, Theorem 19 and Corrolary 21], if $G$ is a direct sum
of cyclic $p$-groups such that $f_{n}(G)=\gamma$, $n\in\mathbb{N}_{0}$, for
some cardinal $\gamma$, then $G$ is a weakly IF-group. Letting
$\gamma\geq\aleph_{0}$, then Proposition 2.16 (1) tells us that every
characteristic subgroup of $G$ is fully invariant, so that $G$ is a weakly IC-
group. Suppose $A$ is such a direct summand of $G$ that its Ulm-Kaplansky
invariants have the properties $f_{0}(A)\geq 2$, provided $f_{0}(A)\neq 0$,
and all $f_{n}(A)$ form an increasing sequence of non-negative integers. Thus
we can apply [6, Corrolary 18] to get that $A$ is not a weakly $IF$-group, and
hence the posed restrictions on $f_{n}(A)$ guarantee that $A$ is also not an
weakly IC-group.

Observe that the following technicality is similar but independent to Theorem
2.8 from [1].

###### Lemma 2.27

Let $B=\bigoplus_{k\geq 1}B_{k}$, where $B_{k}=\bigoplus\mathbb{Z}(p^{k})$ and
$\pi_{k}:B\rightarrow B_{k}$ are the corresponding projections, $H$ is a
characteristic subgroup of $B$. Then $\pi_{k}(H)=p^{n_{k}}B_{k}$, where:

(1) $n_{k}\leq k$ for all $k\geq 1$;

(2) $n_{k}\leq n_{k+r}\leq n_{k}+r$ for all $k,r\geq 1$;

(3) if $n_{k}<k$, then $p^{n_{k}+r}B_{k+r}\leq H$ for all $k,r\geq 1$;

(4) if $r(B_{k})\geq 2$, then $\pi_{k}(H)\leq H$.

Proof. Assume that $\pi_{k}(H)\neq\\{0\\}$. If $r(B_{k})=1$, then
$\pi_{k}(H)=p^{n_{k}}B_{k}$ for some $n_{k}<k$. Let $r(B_{k})>1$,
$B_{k}=\bigoplus_{i\in I}\langle a_{i}\rangle$ and
$\theta_{i}:B\rightarrow\langle a_{i}\rangle$ are the corresponding
projections. Thus, $\theta_{i}(H)=\langle p^{n_{k_{i}}}a_{i}\rangle$. Let
$n_{k}=\mathop{\rm min}\nolimits\\{n_{k_{i}}\,|\,i\in I\\}$. If
$f:a_{k}\rightarrow a_{i}$ is an isomorphism, then
$p^{n_{k}}a_{i}=f(p^{n_{k}}a_{k})\in H$ in virtue of Lemma 2.15. So, in this
case, $\pi_{k}(H)=p^{n_{k}}B_{k}\leq H$, i.e., points (1) and (4) are
fulfilled.

(2) Let $x\in p^{n_{k}}B_{k}\setminus p^{n_{k}+1}B_{k}$. Thus
$o(x)=p^{k-n_{k}}$, so $f(x)\in p^{n_{k}+r}B_{k+r}\leq H$ for every
$f\in\mathrm{Hom}\,(B_{k},B_{k+r})$. Therefore, $n_{k+r}\leq n_{k}+r$. If
$y\in H\cap p^{n_{k+r}}B_{k+r}$, where $n_{k+r}<n_{k}$, then $\varphi(y)\in
p^{n_{k+r}}B_{k}$ for some $\varphi\in\mathrm{Hom}\,(B_{k+r},B_{k})$, because
$\varphi(y)\in H$ contradicts $\pi_{k}(H)=p^{n_{k}}B_{k}$. Finally, the
inequalities $n_{k}\leq n_{k+r}\leq n_{k}+r$ hold.

(3) The proof is analogous to the previous point. In fact, if
$B_{k}=\bigoplus_{i\in I}\langle a_{i}\rangle$, then for every $x\in
p^{n_{k}+r}B_{k+r}$ such that $n_{k}<k$ there is
$f\in\mathrm{Hom}\,(B_{n_{k}},B_{k+r})$ with $f(p^{n_{k}}a_{i})=x$. Applying
now Lemma 2.15, we conclude that $x\in H$, i.e., $p^{n_{k}+r}B_{k+r}\leq H$,
as wanted.

The next statement is often attributed to Kaplansky who showed it for modules
(cf. [8, Theorem 26, p. 63]).

###### Proposition 2.28

If $G=\bigoplus_{i\geq 1}G_{i}$ is a $2$-group, where
$G_{i}=\bigoplus\mathbb{Z}(2^{i})$ is a direct sum of cyclic groups, then each
characteristic subgroup of $G$ is fully invariant if, and only if, $G$ has at
most two Ulm invariants equal to one, and if it has exactly two, they
correspond to successive ordinals.

Proof. ”$\Rightarrow$”. Assume that $f_{k-1}(G)=f_{k+r-1}(G)=1$, where $r>1$,
and $\langle a\rangle\oplus\langle b\rangle$ is the corresponding direct
summand of $G$ with $o(a)=2^{k}$, $o(b)=2^{k+r}$. Suppose
$h=2^{k-1}a+2^{k+r-2}b$, $H=\langle h\rangle=\\{0,2^{k-1}a\pm
2^{k+r-2}b,2^{k+r-1}b\\}$ and

$\underline{H}=H\oplus\left(\bigoplus_{k+1\leq i\leq
k+r-1}G_{i}\right)[2]\oplus\left(\bigoplus_{i\geq k+r+1}G_{i}\right)[2^{2}].$

Thus, one verifies that the subgroup $\underline{H}$ has to be characteristic
in $G$. Indeed, $H$ is characteristic in $\langle a\rangle\oplus\langle
b\rangle$, $f(\underline{H})=\\{0\\}$ for $f\in\mathrm{Hom}\,(\bigoplus_{i\geq
k}G_{i},\bigoplus_{i<k}G_{i})$, $\varphi(H)\leq\underline{H}$ for
$\varphi\in\mathrm{Hom}\,\bigl{(}\langle a\rangle\oplus\langle
b\rangle,\bigoplus_{i>k,\,i\neq k+r}G_{i}\bigr{)}$ and
$\bigl{(}\bigoplus_{k+1\leq i\leq
k+r-1}G_{i}\bigr{)}[2]\oplus\bigl{(}\bigoplus_{i\geq
k+r+1}G_{i}\bigr{)}[2^{2}]$ is fully invariant in $\bigl{(}\bigoplus_{k+1\leq
i\leq k+r-1}G_{i}\bigr{)}\oplus\bigl{(}\bigoplus_{i\geq k+r+1}G_{i}\bigr{)}$.
But $H$ is not fully invariant since for the projection $\alpha$ of $G$ on
$\langle a\rangle$, we have $\alpha(H)=\langle 2^{k-1}a\rangle\nleq H$, as
expected.

”$\Leftarrow$”. Suppose that $G_{k}=\langle a\rangle$, $G_{k+1}=\langle
b\rangle$ and $r(G_{i})\geq 2$ with $i\neq k,k+1$. Let $\pi:G\rightarrow
G_{k}$, $\theta:G\rightarrow G_{k+1}$ be the corresponding projections and let
$H\leq G$ be a characteristic subgroup of $G$. It follows with the aid of
Lemma 2.15 that if $h=x_{i}+y$, where $x_{i}\in G_{i}$ for some $i\neq k,k+1$,
then $x_{i}\in H$ and $\beta(H)\leq H$, where $\beta$ is the projection of $G$
on $\bigoplus_{j\neq k,\,k+1}G_{j}$, and $\beta(H)$ is fully invariant in $G$.
Letting $\pi(H)=p^{n_{k}}a$, $\theta(H)=p^{n_{k+1}}b$, then $p^{n_{k}+1}b\in
H$. If $p^{n_{k}+1}=p^{n_{k+1}}$, then $p^{n_{k}}a\in H$ since in this case
$p^{n_{k}}a+y\in H$ for some $y\in H$, as it was noted above, and so
$p^{n_{k}}a\in H$. One inspects now that $H$ is really fully invariant in $G$.
If, however, $p^{n_{k}+1}>p^{n_{k+1}}$, then $p^{n_{k+1}}\leq p^{n_{k}}$, but
in this situation the relation $p^{n_{k+1}}a\in H$ is possible only if
$p^{n_{k+1}}=p^{n_{k}}$, hence $\pi(H)\leq H$ which yields $\theta(H)\leq H$,
i.e., $H$ is fully invariant. In the remaining case when
$p^{n_{k}+1}<p^{n_{k+1}}$, it can be inferred that $p^{n_{k}+1}b\in H$ that is
impossible.

We can, alternatively, extend the last result to the following one.

###### Proposition 2.29

Given $G$ is a separable $2$-group, then each characteristic subgroup of $G$
is fully invariant if, and only if, $G$ has at most two Ulm invariants equal
to one, and if it has exactly two, they correspond to successive ordinals,
i.e., if, and only if, the corresponding properties are satisfied by its basic
subgroup.

Proof. ”$\Rightarrow$”. In the same manner, as showed above in the preceding
proposition, if $B=\bigoplus_{i\geq 1}B_{i}$, where
$B_{i}=\bigoplus\mathbb{Z}(2^{i})$, is a basic subgroup in $G$, then we may
write
$G=\left(\bigoplus_{i=1}^{n}B_{i}\right)\oplus\left(B_{n}^{*}+2^{n}G\right)$
for any $n\in\mathbb{N}$, where $B_{n}^{*}=\bigoplus_{i\geq n+1}B_{i}$.

Let us now $f_{k-1}(G)=f_{k+r-1}(G)=1$, where $r>1$, and $\langle
a\rangle\oplus\langle b\rangle$ is the corresponding direct summand of $G$
with $o(a)=2^{k}$, $o(b)=2^{k+r}$. Suppose $h=2^{k-1}a+2^{k+r-2}b$, $H=\langle
h\rangle=\\{0,2^{k-1}a\pm 2^{k+r-2}b,2^{k+r-1}b\\}$ and

$\underline{H}=H\oplus\left(\bigoplus_{k+1\leq i\leq
k+r-1}B_{i}\right)[2]\oplus\bigl{(}B_{k+r}^{*}+2^{k+r}G\bigr{)}[2^{2}].$

Thus, one inspects that $\underline{H}$ is a characteristic, but not fully
invariant subgroup in $G$. Note that the separability in this case is not
used.

”$\Leftarrow$”. Let $H$ be a characteristic subgroup in $G$, and let $x\in H$
and $y=f(x)$ for some $f\in\mathrm{E}(G)$. Since $G$ is separable, we know
that the elements $x$ and $f(x)$ can be embedded in some basic subgroup, say
$B$, of $G$. Consulting with Lemma 2.34 stated and proved below, one finds
that $H\cap B$ is characteristic in $B$, so it is fully invariant. And since
$\mathrm{h}^{B}(x)\leq\mathrm{h}^{B}(f(x))$ as well as $B$ is fully
transitive, it must be that $\varphi(x)=f(x)$ for some
$\varphi\in\mathrm{E}(B)$, whence $f(x)\in H$, as required.

The next technical claim is rather curious and very helpful for our
objectives.

###### Lemma 2.30

If $G$ is a $p$-group and $H$ is its characteristic subgroup such that the
quotient-group $H/(H\cap p^{\omega}G)$ is unbounded, then $p^{\omega}G\leq H$.

Proof. If $B=\bigoplus_{i\geq 1}B_{i}$, where
$B_{i}=\bigoplus\mathbb{Z}(p^{i})$, is a basic subgroup in $G$, then we can
write
$G=\left(\bigoplus_{i=1}^{n}B_{i}\right)\oplus\left(B_{n}^{*}+p^{n}G\right)$
for each $n\in\mathbb{N}$, where $B_{n}^{*}=\bigoplus_{i\geq n+1}B_{i}$. Let
$\pi_{n}:G\rightarrow B_{1}\oplus\dots\oplus B_{n}$ be the corresponding
projections. Since the factor-group $H/(H\cap p^{\omega}G)$ is unbounded, for
every $s\in\mathbb{N}$, there exist elements $x=p^{s}h$, $h\in H$ with
$x\notin p^{\omega}G$. Note that $p^{s}\pi_{n_{s}}(h)\neq 0$ for some
$n_{s}>s$ as for otherwise we will have $x\in B_{n_{s}}^{*}+p^{n_{s}}G$ for
each $n_{s}>s$. If now $o(h)=p^{m_{s}}$, then $m_{s}\geq s$. By assumption, we
get $x\in B_{k+m_{s}}^{*}+p^{k+m_{s}}G$ for all $k\geq 1$, so $x\in p^{k+1}G$,
i.e., $x\in p^{\omega}G$, thus contradicting the choice of the element $x$.
Therefore, for every natural $t$, there will exist $n_{t}\in\mathbb{N}$ and
$x\in H$ such that $o(\pi_{n_{t}}(x))\geq t$. But since $U(y)\geq
U(\pi_{n_{t}}(x))$ for each $y\in p^{\omega}G$ with $o(y)\leq t$, we derive
that $f(\pi_{n_{t}}(x))$ for some $f\in\mathrm{Hom}(B_{1}\oplus\dots\oplus
B_{n_{t}},B_{n_{t}}^{*}+p^{n_{t}}G)$. Thus, it follows from Lemma 2.15 that
$y\in H$. Finally, $p^{\omega}G\leq H$, as formulated.

The following assertion is closely related to the previous one.

###### Proposition 2.31

If $G$ is a $p$-group such that either $f_{n}(G)=0$ or $f_{n}(G)\geq 2$ for
all $n\in\mathbb{N}\cup\\{0\\}$, then, for every characteristic subgroup $H$
of $G$, the subgroup $H^{\prime}=H+p^{\omega}G$ is fully invariant in $G$. In
particular, if $H/(H\cap p^{\omega}G)$ is unbounded, then $H$ is fully
invariant in $G$.

Proof. Having in mind Proposition 2.29, the quotient $H^{\prime}/p^{\omega}G$
is fully invariant in $G/p^{\omega}G$, and consequently $H^{\prime}$ is fully
invariant in $G$, as stated. The second part is now obvious by using the
argumentation from the previous statement.

To avoid any further misunderstanding and confusion, let us recall that
$p^{n}G[p^{m}]$ just means $(p^{n}G)[p^{m}]$ written without brackets for
simpleness.

The next four technical assertions are pivotal for obtaining our chief
results. They somewhat expand closely related results from [3] and [4] (see
also the existing bibliography therewith).

###### Lemma 2.32

If $B$ is a basic subgroup of a reduced $p$-group $G$, then the following two
points are valid:

(1) $G[p^{m}]\leq B+p^{n}G[p^{m}]$ for all $m,n\in\mathbb{N}$;

(2) $G=B+\sum_{i\geq 1}p^{n_{i}}G[p^{m_{i}}]$ for any sequences of natural
numbers $1\leq m_{1}<m_{2}<\dots$ and for all set of natural numbers
$\\{n_{i}\\}$.

Proof. (1) Write $G=B+p^{n}G$ for all $n\in\mathbb{N}$. If now $p^{m}g=0$ for
some $g\in G$, then for $g=b+p^{n}g^{\prime}$, $g^{\prime}\in G$, we have
$p^{m}b+p^{n+m}g^{\prime}=0$. Since $B$ is pure in $G$, it follows that
$p^{m}b=p^{n+m}b^{\prime}$ for some $b^{\prime}\in B$. So,
$p^{n+m}g^{\prime}=-p^{n+m}b^{\prime}$ and, consequently,
$g^{\prime}+b^{\prime}\in G[p^{n+m}]$. Thus,
$p^{n}(g^{\prime}+b^{\prime})=g^{\prime\prime}$ for some $g^{\prime\prime}\in
p^{n}G[p^{m}]$, whence $g=b-p^{n}b^{\prime}+g^{\prime\prime}\in
B+p^{n}G[p^{m}]$, as desired.

(2) follows from (1) and the fact that $G=\sum_{i\geq 1}G[p^{m_{i}}]$ for
every sequences of natural numbers $1\leq m_{1}<m_{2}<\cdots$.

###### Lemma 2.33

If $H$ is a non-bounded characteristic subgroup of a separable $p$-group $G$,
then $G=H+B$ for every basic subgroup $B$ of $G$.

Proof. Employing Lemma 2.32 it is enough to show that for the sequences of
natural numbers $1\leq m_{1}<m_{2}<\cdots$ the inclusion $\sum_{i\geq
1}p^{n_{i}}G[p^{m_{i}}]\leq H$ holds for some set of natural numbers
$\\{n_{i}\\}$. To show that, for every $n\geq 1$ we have
$G=(B_{1}\oplus\dots\oplus B_{n+1})\oplus(B_{n+1}^{*}+p^{n+1}G)$, where
$B_{n+1}^{*}=\bigoplus_{i\geq n+2}B_{i}$. Let $\pi_{n}:G\rightarrow B_{n}$ are
the corresponding projections. Let $h_{1}\in H$ and $p^{n+1}h_{1}=0$,
$p^{n}h_{1}\neq 0$ for some $n\in\mathbb{N}$. Note that
$\pi_{n+k}(p^{n}h_{1})\neq 0$ for some $k\geq 1$. In fact, if
$\pi_{n+k}(p^{n}h_{1})=0$ for every $k\geq 1$, then
$p^{n}h_{1}\in(B_{n+k}^{*}+p^{n+k}G)[p]$. However, since $p^{n+k}G$ is
essential in $B_{n+k}^{*}+p^{n+k}G$, we infer that
$(B_{n+k}^{*}+p^{n+k}G)[p]\leq p^{n+k}G$. Therefore, $0\neq
p^{n}h_{1}\in\bigcap_{k\geq 1}p^{n+k}G=\\{0\\}$ – contradiction. So,
$0\neq\pi_{n+k}(p^{n_{1}}h)=p^{n}\pi_{n+k}(h_{1})\in B_{n+k}$ for some $k\geq
1$. Suppose now $p^{m}_{1}=o(\pi_{n+k}(h_{1}))$. Since
$U(\pi_{n+k}(h_{1}))\leq U(z)$ for each $z\in
p^{n+k-m_{1}}(B_{n+k}^{*}+p^{n+k}G)[p^{m}_{1}]$, Lemma2.15 assures that
$p^{n+k-m_{1}}(B_{n+k}^{*}+p^{n+k}G)[p^{m}_{1}]\leq H$, and since
$p^{n+k-m_{1}}G=p^{n+k-m_{1}}(B_{n+k-m_{1}}^{*}+p^{n+k-m_{1}}G)$, it must be
that $p^{n_{1}}G[p^{m_{1}}]\leq H$, where $n_{1}=n+k-m_{1}$.

Furthermore, since the subgroup $H$ is unbounded, there will exist a natural
number $m_{2}>m_{1}$ and some positive integer $n_{2}$ satisfying the relation
$p^{n_{2}}G[p^{m_{2}}]\leq H$, as wanted.

###### Lemma 2.34

If $H$ is an unbounded characteristic subgroup of a separable $p$-group $G$
and $B\leq G$ is a basic, then $H\cap B$ is basic in $H$.

Proof. Our first goal is to demonstrate that $H\cap B$ is pure in $H$. Indeed,
let $px=b$ for some $x\in H$, $b\in H\cap B$. Since $B$ is pure in $G$, we may
write $b=py$ for some $y\in B$. If $h(x)=h(y)$, then $U(x)=U(y)$, and since
$G$ is transitive we write $\alpha(x)=y$ for some $\alpha\in\mathrm{Aut}(G)$;
so, $y\in H\cap B$. Assuming that $m=h(x)<h(y)$, we then have $h(px)>m+1$ and
thus $f_{m}(G)\neq 0$; consequently, $B_{m+1}\neq\\{0\\}$. Letting $z\in
B_{m+1}[p]$, since $h(z)=m$, we deduce $h(y+z)=h(z)=h(x)$, and so
$U(x)=U(y+z)$; therefore, as above, $y+z\in H\cap B$, where $p(y+z)=b$. Let us
now $h(x)>h(y)=m$. So, $B_{m+1}\neq\\{0\\}$ and there exists $z\in B_{m+1}[p]$
such that $h(y-z)>h(y)$. If, however, $h(x)>h(y-z)$, we similarly can choose
$z_{1}\in B[p]$ with $h(y-z-z_{1})>h(y-z)$. Furthermore, after a finite number
of steps, we get either $h(x)=h(y+u)$, or $h(x)<h(y+u)$ for some $u\in B[p]$.
In both cases, as shown above, the equation $px=b$ is resolvable in $H\cap B$.
But since $B+pH$ also equals to $G$, it follows that $H\cap p^{n}H=p^{n}(H\cap
B)$ for every $n$, i.e., $H\cap B$ is pure in $H$. And since $(B+H)/B\cong
H/(H\cap B)$, where $B+H=G$, the subgroup $H\cap B$ is dense in $H$, i.e.,
$H\cap B$ is basic in $H$, as claimed.

###### Lemma 2.35

If $H$ is a characteristic subgroup of a separable $p$-group $G$ and $B$ is
basic in $G$, then $H\cap B$ is characteristic in $B$.

Proof. For every $n$, we have that $G=(B_{1}\oplus\dots\oplus
B_{n})\oplus(B_{n}^{*}+p^{n}G)$. Let $\pi_{n}:G\rightarrow B_{n}$ are the
corresponding projections. If $H\cap B\neq\\{0\\}$, then
$\pi_{n}(H)\neq\\{0\\}$ for some $n$. If $r(B_{n})>1$, then consulting with
Proposition 2.16 we inspect that $\pi_{n}(H)\leq H$ is fully invariant in
$B_{n}$. Moreover, Lemma 2.15 tells us that $f(H\cap B_{n}),\varphi(H\cap
B_{n})\leq H$ for each $f\in\mathrm{Hom}\,(B_{n},B_{1}\oplus\dots\oplus
B_{n-1})$ and $\varphi\in\mathrm{Hom}\,(B_{n},B_{n}^{*}+p^{n}G)$.

Let us now $x\in H\cap B$ and $x=x_{1}+\dots+x_{m}$, where
$x_{j}=\pi_{i_{j}}(x)\in B_{i_{j}}$. In case that $r(B_{i_{j}})>1$, we can
observe that $\pi_{i_{j}}(H)\leq H$ and that $\pi_{i_{j}}(H)$ is fully
invariant in $B_{i_{j}}$, so we may assume that $r(B_{i_{j}})=1$ for all
$j=1,\dots,m$. Given $\alpha\in\mathrm{Aut}(B)$, we note that
$\pi_{i_{j}}\alpha$ is an automorphism for $B_{i_{j}}$ for all $j=1,\dots,m$.
In fact, if $\pi_{i_{j}}\alpha(y)=0$ for some $0\neq y\in B_{i_{j}}[p]$, then
$\pi_{k}\alpha(y)=0$ for every $k<i_{j}$ and so $0\neq\alpha
y\in(B_{i_{j}}^{*})[p]$. But, in this case, $U(\alpha y)>U(y)$ which is
obviously imposable for getting automorphisms. So,
$\ker(\pi_{i_{j}}\alpha)=0$, and in view of the finite number of the
$B_{i_{j}}$s, it follows that
$(\pi_{i_{j}}\alpha)\\!\upharpoonright\\!B_{i_{j}}$ is an automorphism for all
$j=1,\dots,m$. Consequently,

$\mu=(\pi_{i_{1}}+\dots+\pi_{i_{m}})\alpha\\!\upharpoonright\\!\bigoplus_{j=1}^{m}B_{i_{j}}$

is an automorphism of $\bigoplus_{j=1}^{m}B_{i_{j}}$. Since
$\bigoplus_{j=1}^{m}B_{i_{j}}$ is a direct summand of $B$, $\mu$ can be
considered as an extension of the automorphism of $B$, so
$(\pi_{i_{1}}+\dots+\pi_{i_{m}})\alpha(x)\in H\cap B$, as promised, because it
follows from Lemma 2.15 that
$(1_{B}-(\pi_{i_{1}}+\dots+\pi_{i_{m}}))\alpha(x)\in H\cap B$, as asked for.

We are now in a position to prove one of our main results. It is worthwhile
noticing that the following theorem is similar to Theorem 14 in [5] proved for
IF-groups.

###### Theorem 2.36

The separable $p$-group $G$ is not a weakly IC-group if some of its basic
subgroups $B$ is not a weakly IC-group.

Proof. If $G$ is bounded, then its basic subgroups coincide with $G$, hence
Proposition 2.20 applies to get that $G$ is not a weakly IC-group, as
asserted.

Let us now $G$ be unbounded, and assume in a way of contradiction that $G$ is
a weakly IC-group. Then there exists a characteristic subgroup $H\lneqq G$
such that $H\cong G$. Since $H$ is unbounded, one derives by Lemma 2.34 that
$H\cap B$ is basic subgroup in $H$. If $H\cap B=B$, then $B\leq H$, and since
$H+B=G$, the application of Lemma 2.33 guarantees that $H=G$, that is a
contrary to our initial choice. So, $H\cap B\neq B$. Since $H\cong G$, its
basic subgroups are also isomorphic, say $H\cap B\cong B$ (see, for instance,
[3]). However, Lemma 2.35 ensures that $H\cap B$ is characteristic in $B$,
which is an absurd, so we get the pursued contradiction.

The next technical claim is the key in proving the opposite direction of the
last theorem.

###### Lemma 2.37

(1) If $H$ is an unbounded characteristic subgroup in a fixed basic subgroup
$B$ of a torsion-complete $p$-group $G$, then there exists such a
characteristic subgroup $H^{*}$ in $G$ that $B\cap H^{*}=H$ and $H$ is basic
in $H^{*}$.

(2) If $H$ is an unbounded characteristic subgroup of a separable $p$-group
$G$ and $B$ is basic in $G$, then $B\cap H$ is an unbounded characteristic
subgroup in $B$ and $B\cap H$ is basic in $H$.

Proof. (1) Let $H^{*}=\sum_{f\in\mathrm{Aut}(G)}f(H)$ and $x\in B\cap H^{*}$.
Then $x=y_{1}+\dots+y_{n}$, where $y_{i}=f_{i}(x_{i})$,
$f_{i}\in\mathrm{Aut}(G)$, $x_{i}\in H$; $i=1,\dots,n$. Since $G$ is
separable, the elements $x,y_{1},\dots,y_{n}$ can be included in some basic
subgroup $B^{\prime}$ of $G$ (see, e.g., [4]). Let $\varphi:B\rightarrow
B^{\prime}$ be an isomorphism. However, because $G$ is torsion-complete, then
$\varphi$ could be considered as an automorphism of $G$ (cf. [4]). Therefore,
$z=\varphi^{-1}(x)$, $z_{i}=\varphi^{-1}(y_{i})\in B$ and $U(x)=U(z)$,
$U(z_{i})=U(x_{i})$. Since $z_{i}=\psi_{i}(x_{i})$ for some
$\psi_{i}\in\mathrm{Aut}(B)$, we get that $z_{i}\in H$. So,
$z=z_{1}+\dots+z_{n}\in H$. Furthermore, since $U(z)=U(x)$, it must be that
$x\in H$. But Lemma 2.34 yields that $B\cap H^{*}=H$ is basic in $H^{*}$, as
expected.

(2) Again the application of Lemma 2.34 is a guarantor that we just need to
show that $B\cap H$ is an unbounded characteristic subgroup in $B$. But, to
that purpose, we observe that $B\cap H$ is really unbounded, because $H$ is
unbounded and $B\cap H$ is basic in $H$. On the other hand, $B\cap H$ is
characteristic in $B$ owing to Lemma 2.35, as promised.

And so, we arrive at the next crucial result which, actually, is the converse
of Theorem 2.36 in the case of torsion-complete groups. Specifically, the
following assertion is true:

###### Theorem 2.38

A torsion-complete $p$-group $G$ is a weakly IC-group if, and only if, some
(and hence all) of its basic subgroups $B$ is (are) a weakly IC-group.

Proof. Referring to Theorem 2.36, one observes that it is necessary to prove
only the necessity, because we already established the sufficiency. However,
we shall demonstrate below slightly different arguments to show once again its
validity.

To prove necessity, let $H$ be characteristic in $G$ with $H\neq G$. Since $G$
is a weakly IC-group and $H\cong G$, we deduce that $H$ as well as $G$ is
unbounded. But then $H\cap B$ is characteristic in $B$ and $H\cap B$ is basic
in $H$ in accordance with Lemma 2.37 (2). Note that $H\cap B\neq B$, because
if we assume that $B\leq H$, then in view of the equality $B+H=G$ it would
follow that $H=G$ \- a contradiction. But since $H\cong G$, it follows that
their basic subgroups have to be isomorphic (see, for instance, [3]), say
$B\cap H\cong B$, i.e., $B$ is also a weakly IC-group, as formulated.

To show now the truthfulness of sufficiency, let us assume that $G$ be a
torsion-complete $p$-group and $B$ its basic subgroup which is a weakly IC-
group. According to Proposition 2.20, $B$ is an unbounded group and,
therefore, $G$ is also unbounded. Since $B$ is a weakly IC-group, there exists
a proper characteristic subgroup $S$ of the group $B$ such that $B\cong S$.
Owing to Lemma 2.37, there exists a proper characteristic subgroup $S^{*}$ of
$G$ such that $S^{*}\cap B=S$ and $S$ is a basic subgroup of the group
$S^{*}$. If the quotient $D/S^{*}$ is the divisible part of the quotient
$G/S^{*}$, then a routine verification shows that $D$ is also a characteristic
subgroup of $G$ and $S$ is a basic subgroup in $D$. Since the factor-group
$G/D$ is reduced, it follows that $D$ is a torsion-complete group (see, e.g.,
[3, Corrolary 68.7]). Thus, we obtain that the group $G$ has a proper
characteristic subgroup $D$ such that the basic subgroup $B$ of the group $G$
is isomorphic to the basic subgroup $S$ of the group $D$. Finally, as both $G$
and $D$ are torsion-complete groups, we conclude with the aid of volume II
from [3] that $G\cong D$, i.e., $G$ is a weakly IC-group, as required.

The following commentaries are worth noticing.

The property ”torsion-completeness” plays a key if not facilitating role in
proving the last statement. However, a problem which automatically arises is
of whether or not the theorem remains true by replacing the requirement ”a
torsion-complete $p$-group” by ”a separable $p$-group”, that is, is the
statement of Theorem 2.36 reversible? A logical reason for asking this is
that, in view of volume II of [3], we know that any separable $p$-group can be
embedded as a pure and dense subgroup of a torsion-complete $p$-group. We,
however, conjecture that the answer will be negative.

##  3 Concluding Discussion and Open Problems

We conclude this final section with some valuable comments on the obtained
results.

And so, we firstly will comment the Abelian groups with an isomorphic proper
characteristic subgroup, which groups were termed here as weakly IC-groups.
Some of the analogies and differences with the classes of IC-groups and
strongly IC-groups, both introduced in [2], are as follows: (1) These three
classes are definitely not closed under taking direct summands; (2) They have
totally different structure as shown above.

These discrepancies help us to understand more completely the complicated
situations in these three possibilities.

We close our work with two questions of interest and importance.

In regard to Examples 2.14 and 2.26, we pose the following left query.

Problem 1. Find necessary (and sufficient) conditions under which a direct
summand of an IC-group is again an IC-group. Same holds and for
strongly/weakly IC-groups, respectively.

In connection with Corollary 2.23, we state the following left question.

Problem 2. Do there exist (strongly, weakly) IC-groups which are not (fully)
transitive?

Funding: The scientific work of Andrey R. Chekhlov was supported by the
Ministry of Science and Higher Education of Russia (agreement No.
075-02-2022-884). The scientific work of Peter V. Danchev was partially
supported by the Bulgarian National Science Fund under Grant KP-06 No 32/1 of
December 07, 2019 and by the Junta de Andalucía, FQM 264.

## References

  * [1] K. M. Benabdallah, B. J. Eisenstadt, J. M. Irwin and E. W. Poluianov, The structure of large subgroups of primary Abelian groups, Acta Math. Acad. Scient. Hung. (3-4) 21 (1970), 421–435.
  * [2] A. R. Chekhlov and P. V. Danchev, On abelian groups having all proper fully invariant subgroups isomorphic, Commun. Algebra (12) 43 (2015), 5059–5073.
  * [3] L. Fuchs, Infinite Abelian Groups, Vol. I and Vol. II, Academic Press, New York and London, 1970 and 1973.
  * [4] L. Fuchs, Abelian Groups, Springer (Cham, Switzerland, 2015).
  * [5] S. Ya. Grinshpon and M. M. Nikolskaya, Primary IF-groups, Tomsk State Univ. J. Math. & Mech. (3) 15 (2011), 25–31.
  * [6] S. Ya. Grinshpon and M. M. Nikolskaya (Savinkova), Fully invariant subgroups of Abelian $p$-groups with finite Ulm-Kaplansky invariants, Commun. Algebra (11) 39 (2011), 4273–4282.
  * [7] S. Ya. Grinshpon and M. M. Nikolskaya, Proper fully invariant subgroups of a torsion-free group isomorphic to the group, Tomsk State Univ. J. Math. & Mech. (1) 17 (2012), 11–15.
  * [8] I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1954 and 1969.
  * [9] P. A. Krylov, A. V. Mikhalev and A. A. Tuganbaev, Endomorphism Rings of Abelian Groups, Dordrecht, Boston and London: Kluwer Acad. Publ. (2003).
  * [10] R.S. Pierce, Homomorphisms of Primary Abelian Groups, Topics in Abelian Groups, Chicago, Illinois (1963), 215–310.


[name=Theorem,numberwithin=section]thm
[name=Lemma,numberwithin=section,numberlike=thm]lem
[name=Corollary,numberwithin=section,numberlike=thm]cor
[name=Proposition,numberwithin=section,numberlike=thm]prop
[name=Assumption,numberlike=thm]ass

# Cosmology from Galaxy Redshift Surveys
with PointNet

Sotirios Anagnostidis∗1a Arne Thomsen∗2b Tomasz Kacprzak∗2,3 Tilman Tröster∗2
Luca Biggio1,4 Alexandre Refregier2 Thomas Hofmann1
1Department of Computer Science, ETH Zürich, CH 2Department of Physics, ETH
Zürich, CH
3Swiss Data Science Center, Paul Scherrer Institute, CH 4CSEM SA, Alpnach, CH
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In recent years, deep learning approaches have achieved state-of-the-art
results in the analysis of point cloud data. In cosmology, galaxy redshift
surveys resemble such a permutation invariant collection of positions in
space. These surveys have so far mostly been analysed with two-point
statistics, such as power spectra and correlation functions. The usage of
these summary statistics is best justified on large scales, where the density
field is linear and Gaussian. However, in light of the increased precision
expected from upcoming surveys, the analysis of – intrinsically non-Gaussian –
small angular separations represents an appealing avenue to better constrain
cosmological parameters. In this work, we aim to improve upon two-point
statistics by employing a PointNet-like neural network to regress the values
of the cosmological parameters directly from point cloud data. Our
implementation of PointNets can analyse inputs of
$\mathcal{O}(10^{4})-\mathcal{O}(10^{5})$ galaxies at a time, which improves
upon earlier work for this application by roughly two orders of magnitude.
Additionally, we demonstrate the ability to analyse galaxy redshift survey
data on the lightcone, as opposed to previously static simulation boxes at a
given fixed redshift.

##  1 Introduction

The standard model of cosmology contains a set of free parameters that relate
to fundamental properties of the universe, such as the fraction of dark energy
and dark matter, the dark energy equation of state, and the rate of cosmic
expansion. The matter density distribution evolves under the laws of gravity
and cosmic expansion from a smooth, Gaussian field in the early universe, to a
complex network of halos, filaments, and voids in the late universe. The
spatial distribution of galaxies traces the underlying dark matter density
field. The cosmological parameters can be inferred by comparing the structures
in this distribution from the observed survey data and those predicted by
theoretical models (see Sánchez et al., 2017, and references therein). Current
cosmological analyses typically compress the information contained in the
galaxy positions into two-point statistics, such as power spectra and
correlation functions (Alam et al., 2021). The two-point statistics are only
sufficient for Gaussian random fields and therefore cannot fully capture the
richer information contained in the galaxy positions.

Using more powerful features, such as higher-order correlation functions
(Samushia et al., 2021), or other approaches (Banerjee and Abel, 2021;
Eickenberg et al., 2022), either singularly or combined with each other, can
improve performance. Yet, the selection of an appropriate set of summary
statistics is problem-dependent and prone to human bias. This aspect motivates
the adoption of tools to automatically extract and combine summary statistics
in a meaningful way, i.e. to improve the prediction of cosmological
parameters. Aside from galaxy positions, other features can be added to
increase the constraining power. An example of such a property is the local
density of the galaxy. Marked correlation functions (Satpathy et al., 2019;
Philcox et al., 2021; Massara et al., 2022) report increased precision of
measurements for various strategies to up-weight certain density regions. This
requires careful design of the marking strategy.

Motivated by advancements in machine learning, recent work attempts to
estimate parameters from the galaxy positions directly. As an early example,
Ntampaka et al. (2020) frame the task as a 3D voxel regression, by applying an
arbitrary scale partition of the input space into equally sized blocks. A
disadvantage of this approach is that the dynamical range of the density of
galaxies makes the voxelisation either very lossy or computationally
inefficient. Furthermore, the permutation symmetries of the task are not
respected.

A more natural representation is treating the data as a graph and employing
graph neural networks (GNNs) Villanueva-Domingo and Villaescusa-Navarro
(2022); Makinen et al. (2022), which are able to handle irregular and sparse
data, via a message passing protocol defined between connected nodes. However,
this setting comes with two fundamental limitations. Firstly, treating the
data as a graph requires defining an adjacency matrix between the galaxies a
priori. Neighbors in this case are usually specified as points within a hard-
defined threshold distance. Secondly, these methods can scale poorly when
increasing the available number of points, which is of major concern due to
the relatively large galaxy catalogues which in the ongoing galaxy redshift
surveys like DESI (Flaugher and Bebek, 2014) can reach the order of
$\mathcal{O}(10^{6})$ objects.

We instead propose to treat the positional data directly in 3D space without
making any further assumptions. As a result, the underlying relationship
between the positions can be learned automatically and no assumptions on an
initial adjacency matrix are needed. Point clouds are a natural representation
of these geometric point sets, where an invariance to the order of the points
within the set is desired. Distances between points define a local
neighborhood, whose properties should give good indications of the underlying
data distribution. We base our work on PointNet (Qi et al., 2016) and
subsequent advances that learn features from point sets in a metric space.

An additional advantage of the analysis of PointNet is the fact that local
density information can be naturally included in the input vector, extending
the set of 3D positions $(x,y,z)$ with an additional local density proxy
parameter, for example the halo virial mass $M$. Marked correlation functions
use the local density values to optimally weight the correlation, and this
weighting strategy needs to be carefully tuned. In contrast, the PointNet
machine learning algorithm automatically finds the optimal weighting to
maximize the information gain.

In this work, we demonstrate the potential of using PointNets for analysing
redshift surveys with a large number of galaxies, reaching
$\mathcal{O}(10^{4})-\mathcal{O}(10^{5})$ objects, with prospects of further
increase when ran on larger hardware. We show that the local density
information can be straightforwardly included in the data vector, extending
the set of galaxy properties by one additional parameter. We compare the
accuracy of PointNets and two-point statistics in predicting the underlying
cosmological parameters and show that our approach performs favourably,
especially when the additional feature $M$ is included.

##  2 Methodology

Figure 1: Schematic illustration of the proposed approach. For galaxy redshift
surveys, cosmological parameter inference is traditionally performed by
manually extracting summary statistics, i.e. features, from a set of points in
3D coordinates. Our approach differs in that relevant features are
automatically extracted by a hierarchical PointNet directly processing the
point cloud and outputting the corresponding cosmological parameters. This
way, the need for introducing hand-crafted summary statistics can be entirely
avoided.

Our dataset is derived from the dark matter halo catalogs contained in the
CosmoGridV1 (Kacprzak et al., 2022; Fluri et al., 2022), which contains dark-
matter-only N-body simulations varying the six cosmological parameters
$\Omega_{m}$, $\sigma_{8}$, $w_{0}$, $H_{0}$, $n_{s}$ and $\Omega_{b}$ of the
$w$CDM model. The distribution of halos is primarily affected by the
parameters $\Omega_{m}$, the present-day fraction of matter in the Universe,
and $\sigma_{8}$, a measure of the variance, or clumpiness, of the matter
distribution. We, therefore, restrict the present study to these two
parameters. The remaining ones are implicitly marginalized over in the
analysis. To simplify the physics modeling, we use dark matter halo positions
as a proxy for galaxy positions and work in physical space, rather than
redshift space.

The positions of the halos define a permutation-invariant set of points in 3D
comoving coordinates. We assume that the distance metric between these points
defines a local neighborhood that encodes properties of the underlying data
generation process. This concept of local neighborhood allows for the capture
of attributes that may not be uniformly distributed across the set of
available points, which is in contrast to the global view two-point functions
provide. Therefore, our basic intuition is to exploit the local structure and
learn fine-tuned spatial features based on these neighborhood attributes. In
addition, we include the halo mass $M$ as a proxy for the local density in
selected experiments. Then, the input to the neural net is a set of positions
and mass: $(x,y,z,M)$.

Based on recent work on hierarchical neural network point set learning (Qi et
al., 2017; Zhao et al., 2021), we partition the set of halos on overlapping
local regions by the distance metric of the underlying space, which is
Euclidean. By aggregating predictions across these regions on increasingly
large scales, we progressively expand the receptive field and produce higher-
level features, see also Fig. 1. This enables us to efficiently train on large
sets of point clouds, which capture both local and global information.

##  3 Experiments

Data. The survey volume encompasses a round patch of 1000 square degrees,
which is roughly $2.5\%$ of the full sky and a redshift range of
$z\in[0.4,1]$, see Fig. 2(a) for a visualisation. The dark matter halos create
a lightcone, where the properties of the density field evolve in redshift; in
effect, the higher $z$ halos are clustered the way they were at the time in
cosmic history corresponding to that redshift. The number of dark matter halos
contained in this volume is cosmology-dependent. For consistency, we only
consider point clouds of a fixed number of points (see Table 1) and discard
cosmologies with fewer points, which are only found for the lowest values of
the $\sigma_{8}$ parameter. These are orders of magnitude more points than in
previous analyses employing GNNs, which was $\mathcal{O}(10^{2})$ in Makinen
et al. (2022), and $\mathcal{O}(10^{3})$ in Villanueva-Domingo and
Villaescusa-Navarro (2022).

(a) Example point cloud of dark matter halos on a lightcone, where the size of
the points is proportional to their virialized mass $M$. The axes give the
comoving coordinates in units of Mpc and $1.6\times 10^{4}$ points are
depicted in total. The two subfigures include boxes of size 300 Mpc. There,
the edge-strength corresponds to the inverse of the distance between points.

(b) Comparison of the cosmological constraints. The contours correspond to the
posterior distribution of $\Omega_{m}$, the fraction of matter in the
Universe, and $\sigma_{8}$, the variance of the matter field. PointNeXt
outperforms the two-point statistic for equal numbers of points, which is set
to $3.2\times 10^{4}$.

The CosmoGridV1 is made up of 2500 unique combinations of the six cosmological
parameters distributed according to a Sobol sequence. The suite contains seven
independent N-body simulation runs for each such grid point, which result from
differing random initial conditions and each cover the full sky. We cut out 12
non-overlapping survey patches from each of these to arrive at 84 independent
realizations of the point cloud per cosmology. Therefore, the dataset
encompasses a total of 210 000 elements before the removal of cosmologies
which only contain an insufficient number of halos. We split the original data
into training ($90\%$) and testing ($10\%$).

We obtain consistent comoving coordinates throughout the grid of cosmologies
by fixing a fiducial cosmology with respect to which the observable angular
and redshift coordinates are converted. In addition to the comoving
coordinates $(x,y,z)$, the catalogs contain the halo mass $M$.

Point cloud networks. We build upon the recent innovations in point cloud
processing by neural networks in the PointNet family (Qi et al., 2016). These
are based on the idea that the density of point sets is not uniform across the
receptive field, hindering performance when methods disregard this fact.
PointNet++ (Qi et al., 2017) and succeeding improvements proposed in PointNeXt
(Qian et al., 2022), learn layers in an adaptive fashion, from multiple
scales. We adopt the latter of these methods, which was recently shown to
achieve state-of-the-art results across a variety of point cloud applications.

Two-point statistics. We use isotropic two-point correlation functions as the
baseline for our experiments and estimate them using the TreeCorr Jarvis
(2015) library. The correlation functions are calculated between the same
numbers of points as the ones that make up the point clouds. We compress these
two-point feature vectors using a simple multilayer perceptron (MLP)
regressor, allowing for a more direct comparison to the PointNeXt models, as
well as enabling efficient conditional density estimation. For further
details, see Appendices A.2 and A.3.

# points | PointNeXt (pos) | PointNeXt (pos+$M$) | Two-point statistic
---|---|---|---
$0.8\times 10^{4}$ | $3.6\times 10^{-3}$ | $1.3\times 10^{-3}$ | $8.3\times 10^{-3}$
$1.6\times 10^{4}$ | $2.4\times 10^{-3}$ | $6.7\times 10^{-4}$ | $3.4\times 10^{-3}$
$3.2\times 10^{4}$ | $1.3\times 10^{-3}$ | $5.8\times 10^{-4}$ | $1.8\times 10^{-3}$
Table 1: Comparison of the MSE for our PointNeXt and an MLP trained on two-
point correlations. The MSE is calculated on a test set of predictions for
both $\Omega_{m}$ and $\sigma_{8}$. The results reported in the third column
are obtained with the halo mass as a feature additional to their position.

Results. We consider two metrics to assess the performance of our methods: the
mean-squared error (MSE) on the test set and the posterior distribution of the
cosmological parameters with respect to a fixed mock observation.

The MSE of PointNeXt and the two-point statistic is shown in Table 1. The
results indicate that PointNeXt obtains more precise predictions for the same
number of halos. The difference between the two methods is larger for smaller
numbers of points, which implies that the PointNets are more robust to noise.
More striking is the boost in performance the inclusion of the mass $M$ as an
additional feature brings. For the PointNets, the MSE drops by more than a
factor of two with this additional information.

To estimate the posterior distributions of cosmological parameters, we perform
conditional density estimation on the regressed parameters from PointNeXt and
the compressor of the two-point statistics. We refer the reader to appendix
A.3 for details on the posterior inference. In Fig. 2(b), we visualize the
resulting parameter posteriors when using $3.2\times 10^{4}$ points. In line
with what is observed for the MSE, PointNeXt is able to constrain the
parameters more precisely than the two-point statistic, especially the
parameter $\sigma_{8}$, which describes the variance of the matter field.
While the MSE of the PointNeXt model is markedly improved by the inclusion of
the mass $M$ as a feature, we find that this is mainly due to a reduction of
outliers. The size of the $(\Omega_{m},\sigma_{8})$ posterior is however not
strongly affected by the inclusion of the halo mass as a feature.

##  4 Discussion

For our experimental setting, the standard two-point correlation function does
not offer any further potential for improvement beyond the results reported in
Table 1, thus reaching its limits. While marked functions could be used to
incorporate the evidently beneficial mass information into the correlators,
these require careful modeling choices. In contrast, the PointNets
automatically learn how to optimally handle the information contained in this
additional feature, which makes them especially suited for the task. As in
this study we have not performed an extensive optimization of the PointNets
both in terms of hyperparameter tuning and the selection of architecture, we
anticipate that the gap in performance between the methods in Table 1 can be
increased, even when considering only the galaxy positions. Similarly, the
size and complexity of the PointNet can be increased on larger hardware, thus
potentially leading to more precise parameter measurements. Another avenue of
improvement is to utilize models that ensure the adherence to physical
symmetries like rotation or translation equivariance, as this was shown to
improve the results in Makinen et al. (2022).

##  5 Conclusion and Outlook

In this work, we use PointNets to directly regress cosmological parameters
from simplified simulated galaxy redshift surveys. We show that for an equal
number of points, our PointNet implementation outperforms the de facto
standard approach, the two-point correlation function. We show that the
improvement is twofold: Firstly, the PointNets obtain better results when
acting on the positions only, presumably by extracting non-Gaussian
information the correlators aren’t sensitive to. Secondly, the inclusion of
the mass as an additional feature improves their performance in terms of the
MSE, while only marignally improving the constraints. Given the ease of
including more features to the point vector, this is a promising result. We
leave the investigation of how it can be used to gain constraining power to
future work.

With this in mind, it is a notable advantage of the PointNets that additional
features can be incorporated directly and without the need for fine-tuning,
which is not the case for extensions to the correlation function. Therefore, a
promising direction for follow up work is to apply the method to point clouds
made up of galaxies instead of dark matter halos, as more physical features
apart from the positions and mass can be modeled. For example, the inclusion
of measurements of galaxy magnitudes and colors can help constrain galaxy
evolution models in connection to their density environment.

Our implementation of PointNets can currently handle
$\mathcal{O}(10^{4})-\mathcal{O}(10^{5})$ galaxies at a time, which marks an
improvement of roughly two orders of magnitude compared to earlier work
deploying deep learning methods to this application. As there is still an
order of magnitude gap to the $\mathcal{O}(10^{6})$ galaxy positions mapped in
ongoing state-of-the-art galaxy redshift surveys, scaling our method to even
larger numbers of points is another direction for future extensions of this
work. We expect that this can be achieved, for example, by using the latest
large memory, multi-GPU nodes.

## Impact statement

Upcoming galaxy surveys are expected to provide a wealth of new data that can
shed light on the nature of dark matter and dark energy. The design of new
tools and techniques to process such data is of paramount importance to this
task. The presented approach contributes to this by demonstrating a
simulation-based inference analysis framework with PointNets. They can
efficiently operate on large galaxy point clouds, naturally incorporate more
galaxy properties, and maximize the information gain form the data. We do not
foresee any ethical concerns with this work.

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##  Appendix A Experimental Setup

###  A.1 Point cloud networks

More details regarding our implementation setup are displayed in Table 2. We
highlight that we mainly follow the experimental settings of Qian et al.
[2022]. For our most computational intensive experiments we train on 8 NVIDIA
Tesla P100 GPUs.

| Hyperparameters | Value
---|---|---
Architecture | radius | 0.1
strides | [4, 4, 4, 4]
blocks | [4, 4, 7, 4]
width | 32
expansion | 4
nsample | 32
Training | Learning rate | $3\epsilon^{-4}$
Learning rate scheduler | ‘cosine’
Adam $(\beta_{1},\beta_{2})$ | (0.9, 0.999)
Batch size | 64
Weight decay | 0.0001
Loss | ‘mse-loss’
|  |
Table 2: Hyperparameters for the PointNeXt experiments.

###  A.2 Two-point statistics

We fix the number of randoms in the Landy-Szalay estimator Landy and Szalay
[1993] to $2.56\times 10^{5}$ in all cases. The summary statistic leaves
freedom to choose appropriate binning, which controls both the scales included
(via the bin limits), and the amount of noise (via the bin widths). We fix
these parameters such that we obtain 50 linearly spaced bins in the range from
0 Mpc to 300 Mpc.

###  A.3 Cosmology parameter inference

We wish to derive posterior distributions on the cosmological parameters,
similar to what is being done in traditional cosmological analyses. Since the
likelihood of the halo distribution is intractable, we make use of simulation-
based inference Alsing et al. [2019], Cranmer et al. [2020]. The high
dimensionality of the input data, the halo catalog, makes the direct
application of conditional density estimation difficult. We therefore compress
the input data to a low-dimensional representation on which we can then
perform conditional neural density estimation. Different choices as to how one
can compress the high-dimensional data have been proposed for cosmological
data sets Charnock et al. [2018], Fluri et al. [2019], Jeffrey et al. [2021],
Fluri et al. [2021]. Here we consider three approaches: using the cosmological
parameters as summaries, together with a simple MSE loss; using the
cosmological parameters as summaries while also predicting their covariance,
together with a likelihood loss, similar to Fluri et al. [2019]; and using
summaries that maximise the mutual information with the true cosmological
parameters, similar to Jeffrey et al. [2021].

We assess the efficiency of these three compression schemes on the two-point
statistics due to the lower computational cost compared to the PointNeXt
model. We use an MLP with two hidden layers of width $n_{\mathrm{hidden}}=128$
as the compressor, trained for 200 epochs, with the learning rate decreasing
from $4\times 10^{-4}$ every 40 epoch by a factor of two. We find that the
differences between these three approaches when compressing the two-point
statistics are negligible, both in terms of the MSE, as well as the inferred
posterior distributions. We therefore use the cosmological parameters as
summaries, together with a MSE loss, as our fiducial choice for both the two-
point data and PointNeXt.

We use a three-component Gaussian mixture model for the conditional density
estimation of the summaries. The posterior distributions are then obtained by
conditioning the density estimator on a single representative sample of the
summaries. The coverage of the estimated posterior distributions are validated
using the empirical expected coverage probability Hermans et al. [2021] and
are found to be well calibrated, as shown in Fig. 3.

As a sanity check we also approximate a Gaussian likelihood for the two-point
statistics by considering the mean over all realisations at each grid point as
the model and using the realisations at the fiducial cosmology for the data
covariance matrix. This gives a noisy and sparse estimate of the likelihood
over the whole prior volume. The resulting posterior distribution agrees with
that of the compressed two-point statistics but we refrain from a quantitative
comparison due to the sparsity of the posterior samples and noise of this
Gaussian likelihood approximation.

Figure 3: Expected coverage of the posteriors in Fig. 2(b). A perfectly
calibrated posterior has an empirical expected coverage probability equal to
the nominal expected coverage probability, corresponding to a diagonal line.
Conservative posteriors lie above the diagonal, while overconfident posteriors
lie below it.