# Normal subgroups of iterated wreath products of symmetric groups and
alternating with symmetric groups

Ruslan V. Skuratovskii

###### Abstract

Normal subgroups and there properties for finite and infinite iterated wreath
products $S_{n_{1}}\wr\ldots\wr S_{n_{m}}$, $n,m\in\mathbb{N}$ and $A_{n}\wr
S_{n}$ are founded. The special classes of normal subgroups and there orders
are investigated. Special classes of normal subgroups are investigated and
their generators are found and presented in the form of Kaloujnine tables.
Further, the monolith (i.e. the unique minimal normal subgroup) of these
wreath products is investigated.

Inverse limit of wreath product of permutation groups is found.

. _Key words_ : normal subgroups, iterated wreath product of symmetric groups,
normal subgroups of wreath product of symmetric group and alternating group.

MSC: 20D10, 20F05, 20B05, 20B25, 20B22, 20B07, 20E08, 20E28, 20B35.

##  1 Introduction

This work is the continuation of the previous researches of the author [1,
27]. Also this research clarifies our previous results from [1]. One of the
modern brunches of group theory researches is a study of isometry of geometry
structures as graph trees and of combinatorial-topological structures [30]. In
many cases it is related with studying of groups of automorphisms and normal
structures of these groups. Isometry group of Baire space of type ($n_{1}$,
$n_{2}$, $\ldots$ ) is isomorphic to spherically homogeneous tree with the
same type. The results were presented on conferences [28, 29].

One of the reasons for the interest in groups acting on trees is that various
interesting examples of groups are naturally depicted as groups acting in
particular on root trees, for example, infinite finitely generated periodic
groups. In addition, groups of automorphisms (isometries) of root trees are
still being actively studied due to the fact that they are quite well arranged
and immerse in them numerous finitely approximate groups containing a number
of subgroups with different properties. Thus, the periodic groups of V.I.
Sushchanskyy and R. I. Grigorchuk are embedded in the groups of automorphisms
of such trees. Thus, it is natural to consider infinitely iterated crown
products of finite groups of substitutions. In particular, the main purpose of
this work is to describe all normal subgroups of the group of automorphisms of
a finite (restricted) regular rooted tree. The paper develops tools for the
study of the normal structure, based on the specifics of the wreath product of
symmetric groups.

A form of commutators of wreath product $A\wr B$ was briefly considered in
[13]. We consider the form of commutator presentation [13] in form of wreath
recursion [7, 11] and as Kaloujnine tableaux, its properties as a normal
subgroup is studied here. We impose our results about of normal subgroups of
permutational wreath product [1].

In this paper we continue the research of subgroups of wreath product which
was stared in [1, 11, 16, 26, 19] in particular in [11] a commutator subgroup
as a normal subgroup of wreath product with non-effective action was
considered by us. Despite this, it was an open question whether the subgroup
of $\underset{i=1}{\overset{k}{\mathop{\wr}}}\,{{\mathcal{S}}_{{{n}_{i}}}}$ is
normal. We generalize the passive group of this wreath product $S_{n}\wr
S_{n}$ to any permutation subgroup $B$ instead of only considering the wreath
product of symmetric groups.

##  2 Preliminaries

Let $G$ be a group. The intersection of all non-trivial normal subgroups
$Mon(G)$ of $G$ is called the monolith of a group $G$. If $Mon(G)\neq<1>$,
then the group $G$ is called monolithic, and, in this case, $Mon(G)$ is the
least non-trivial normal subgroup of $G$.

######  Definition 1.

The unique minimal normal subgroup is called the monolith.

According to [10, 15] by tableaux we call all possible infinite tuple of the
form:

$a=[a_{k}]_{k\in\mathbb{N}},\quad\mbox{where}\quad a_{k}\in
Fun(^{(k-1)}M,G_{k}),\ k\in\mathbb{N}.$ (1)

A group of isometries of the generalized Baire space $E$ [8, 6, 9] over a
family of sets $M_{k}$, $k\in Z$, we denote by $I$ according to [15, 9, 6]. We
suppose that for every $k\in N$ set $M_{k}$ is finite.

A group of isometries $I_{Z_{p}}$ of metric space of integer ring of $p$-adic
numbers is isomorphic as a permutation group to the iterated wreath product of
permutation groups of degree $p$.

######  Definition 2.

We will say that the subset $U<G$ is determined by its $k$-coordinate sets
$[U]_{k}$, $k\in N$, if this subset consists of the most possible tables $a\in
I_{Z_{p}}$ such that for arbitrary $k\in N$ holds $[a]_{k}\in[U]_{k}$.

######  Definition 3.

Let $A$ be some set of the tableaux of form (1), then we will call by the
$k$-projection of the set $A$ the set $[A]_{k}={|a|_{k}:a\in A}$. We call the
set $A$ as $k$-coordinate set if $[A]_{k}=A$ holds.

######  Definition 4.

We denote by $A|_{k}$ a set of all functions $a_{k}$ such that
$[\varepsilon,\ldots,\varepsilon,a_{k},\varepsilon\ldots]\in[A]_{k}$.

Recall that such tableau action on element $a=[a_{k}]_{k\in\mathbb{N}}$ on
element
$m=(m_{k})_{k\in\mathbb{N}}\in M$ is defined by the rule on element:

$m^{a}=(m_{k}^{a_{k}(^{(k-1)}m)})_{k\in\mathbb{N}}.$

The set $G$ of all such tableaux of the form (1) with such action which
equipped by such multiplication rule forms a group. The neutral element is the
table of form $e=[\ldots,\varepsilon,\varepsilon,\ldots]$.

######  Definition 5.

We will say that the subset $U<G$ is determined by its $k$-coordinate sets
$[U]_{k}$, $k\in N$, if this subset consists of the most possible tables $a\in
I$ such that for arbitrary $k\in N$ holds $[a]_{k}\in[U]_{k}$. If $U$ is a
splitting subgroup [15] of the group $I$, which is defined in [15], then
$k$-coordinate subset $[U]_{k}$ are subgroups of the group $U$, and
$[U_{k}]=^{(k)}U\bigcap U^{(k)}$. Besides, family $[U]_{k}$, where $k\in N$
will generate a uniformly dense subgroup $U^{\prime}$ of a group $U$. The
splitting normal subgroups of the group $I$ were also studied in [9].

Let $G$ be a group acting (from the right) by permutations on a set $X$ and
let $H$ be an arbitrary group. Then the (permutational) wreath product $H\wr
G$ is the semidirect product $H^{X}\leftthreetimes G$, where $G$ acts on the
direct power $H^{X}$ by the respective permutations of the direct factors. The
group $C_{p}$ or $(C_{p},X)$ is equipped with a natural action by the left
shift on $X=\\{1,\ldots,p\\}$, $p\in\mathbb{N}$. It is well known that the
wreath product of permutation groups is associative.

The multiplication rule of automorphisms $g$, $h$ which are presented in the
form of wreath recursion [7] $g=(g_{(1)},g_{(2)},\ldots,g_{(d)})\sigma_{g},\
h=(h_{(1)},h_{(2)},\ldots,h_{(d)})\sigma_{h},$ is given by the formula:

$g\cdot
h=(g_{(1)}h_{(\sigma_{g}(1))},g_{(2)}h_{(\sigma_{g}(2))},\ldots,g_{(d)}h_{(\sigma_{g}(d))})\sigma_{g}\sigma_{h}.$

We define $\sigma$ as $(1,2,\ldots,p)$ where $p$ is clear from context.

The set $X^{*}$ is naturally the vertex set of a regular rooted tree, i.e. a
connected graph without cycles and a designated vertex $v_{0}$ called the
root, in which two words are connected by an edge if and only if they are of
form $v$ and $vx$, where $v\in X^{*}$, $x\in X$. The set $X^{n}\subset X^{*}$
is called the $n$-th level of the tree $X^{*}$ and $X^{0}=\\{v_{0}\\}$. Denote
the $i$-th the vertex of $X^{j}$ by $v_{ji}$. Note that the unique vertex
$v_{k,i}$ corresponding to each word $v$ in alphabet $X$. For every
automorphism $g\in Aut{{X}^{*}}$ and every word $v\in X^{*}$ we define the
section (state) $g_{(v)}\in AutX^{*}$ of $g$ at $v$ by the rule:
$g_{(v)}(x)=y$ for $x,y\in X^{*}$ if and only if $g(vx)=g(v)y$. The subtree of
$X^{*}$ induced by the set of vertices $\cup_{i=0}^{k}X^{i}$ is denoted by
$X^{[k]}$. The restriction of the action of an automorphism $g\in AutX^{*}$ to
the subtree $X^{[l]}$ is denoted by $g_{(v)}|_{X^{[l]}}$. The restriction
$g_{(v_{ij})}|_{X^{[1]}}$ is called the vertex permutation (v.p.) of $g$ at
the vertex $v_{ij}$ and denoted by $g_{ij}$. We refer to the endomorphism
$\alpha|_{v}$ restriction of $g$ in a vertex $v$ [7]. For example, if $|X|=2$
then we just have to distinguish active vertices, i.e. the vertices for which
$\alpha|_{v}$ is non-trivial.

We label every vertex of ${{X}^{l}},\,\,\,0\leq l<k$ by sign 0 or 1 depending
on the action of v.p. on it. The resulting vertex-labeled regular tree is an
element of $Aut{{X}^{[k]}}$.

Let us fix some notations. The commutator of two group elements $a$ and $b$ is
denoted by $[a,b]=aba^{-1}b^{-1}.$ We denote by

$a^{b}=bab^{-1}$

the conjugation of $a$ by $b$.

##  3 Normal subgroups of wreath product of symmetric groups

###  3.1 Normal subgroups in $S_{n}\wr S_{m}$

We will notice at once, that further statements and expositions will be true
for $n\geqslant 3$ even in case if degrees of two our symmetric groups from
wreath product are different. Elements of our wreath product $S_{n}\wr
S_{n},n\geqslant 5$, will present in the form of tableaux, where $[a]_{1}$ –
the active element of the table and $a\in S_{n}$,
$[a_{1},a_{2},...,a_{n}]_{2}$ are the passive elements of the table, $a_{i}\in
S_{n}$, viz.:

$[a_{1}]_{1},\ [a_{1},a_{2},\ldots,a_{n}]_{2}.$

The rule of multiplication of elements is described in book of Meldrum [13].
Let us give the following definitions.

######  Definition 6.

The minimal number of transpositions in factorization of permutation
corresponding to element $[a_{i}]_{j}$ on transposition is called a rank of
this permutation, we will denote it by $rnk([a_{i}]_{j})$. We set $rnk(e)=0$.

For example, independent transpositions from the Moore generating system [4]
for $A_{n}$ can be used for factorization of $g$ in product of transpositions.
Note that presentation of $g\in{{A}_{n}}$ always has even number of
transpositions.

Consider the interaction of two interlinked permutations represented by
squares. For this first note that if we have two special types of permutations
$\pi_{1},\pi_{2}\in S_{n}$ that decomposed into the product of transpositions,
of $S_{n}$ then the following formula holds:

$rnk(\pi_{1}\cdot\pi_{2})=rnk(\pi_{1})+rnk(\pi_{2})-2m,m\in\emph{N}.$ (2)

Note that a rank of permutation’s product $\pi_{1}\pi_{2}$ can be lesser then
rank of each permutation $\pi_{1},\pi_{2}$.

######  Definition 7.

The set of elements from $S_{n}\wr S_{n},n\geqslant 5$ of the tableaux form:
$[e]_{1},\ [a_{1},a_{2},\ldots,a_{n}]_{2},$ satisfying the following condition

$\sum_{i=1}^{n}rnk([a_{i}]_{2})=2k,k\in\mathbb{N},$ (3)

we will call set of type $\widetilde{A}$ and denote this set by
$e\wr\widetilde{A}_{n}$.

In other words such subset of element $g$ from $S_{n}\wr S_{n},n\geqslant 5$
belongs to type $\widetilde{A}$ if $g$ corresponds to automorphism of
$X^{[2]}$ having on zero level trivial permutation and vertex permutations on
second level forms an even permutation.

######  Proposition 1.

The set of elements of type $\widetilde{A}^{(0)}$ forms subgroup
$e\wr\widetilde{A_{n}}\lhd S_{n}\wr S_{n}$.

###### Proof.

Let us check $e\wr\widetilde{A_{n}}$ is closed with respect to multiplication
and existing of inverse element.

(1 a) Let $\widetilde{a},\widetilde{b},\widetilde{c}$ are elements of
$e\wr\widetilde{A_{n}}$ presented by tableaux: $[e]_{1},\
[a_{1},a_{2},\ldots,a_{n}]_{2};$
$[e]_{1},\ [b_{1},b_{2},\ldots,b_{n}]_{2};$
$[e]_{1},[c_{1},c_{2},\ldots,c_{n}]_{2}$. Then
$(\widetilde{a}\cdot\widetilde{b})\cdot\widetilde{c}$ can be expressed by
$[e]_{1},\ [(a_{1}\cdot b_{1})\cdot c_{1},(a_{2}\cdot b_{2})\cdot
c_{2},\ldots,(a_{n}\cdot b_{n})\cdot c_{n}]_{2}$, which is equal to,
$[e]_{1},\ [a_{1}\cdot(b_{1}\cdot c_{1}),a_{2}\cdot(b_{2}\cdot
c_{2}),\ldots,a_{n}\cdot(b_{n}\cdot c_{n})]_{2}$ because of associative of
permutation multiplication. This tableau is equal to
$\widetilde{a}\cdot(\widetilde{b}\cdot\widetilde{c})$. So we have that
multiplication is associative.

(1 b) The neutral element $[e]_{1},\ [e,e,\ldots,e]_{2}$.

(1 c) For every element of form $[e]_{1},\ [a_{1},a_{2},\ldots,a_{n}]_{2}$
exist inverse element $\widetilde{a}^{-1}\in e\wr\widetilde{A_{n}}$, of form:
$[e]_{1},\ [a_{1}^{-1},a_{2}^{-1},\ldots,a_{n}^{-1}]_{2}$. Let us note that
$\sum_{i=1}^{n}rnk([a_{i}]_{2})=\sum_{i=1}^{n}rnk([a_{i}^{-1}]_{2})=2k,$
$k\in\mathbb{N}$, since $rnk(a_{i})=rnk(a_{i}^{-1})$, $0\leq i\leq n$.

Therefore $e\wr\widetilde{A_{n}}$ is subgroup in $S_{n}\wr S_{n}$.

Note that for product of permutations $\pi_{1},\pi_{2}\in S_{n}$ following
equation holds

$rnk(\pi_{1}\cdot\pi_{2})=rnk(\pi_{1})+rnk(\pi_{2})-2m,m\in\mathbb{N}.$ (4)

summand ’-2m’ arise because of possible simplifying of transposition in
factorization of permutations $\pi_{1},\pi_{2}\in S_{n}$. Note that rank of
resulting permutation is even.

For proving a normality of $e\wr\widetilde{A_{n}}$ in $S_{n}\wr S_{n}$ we take
an arbitrary element $s\in S_{n}\wr S_{n}$. Note that for $s^{-1}\in S_{n}\wr
S_{n}$ equality of ranks holds
$\sum_{i=1}^{n}rnk([s_{i}]_{2})=\sum_{i=1}^{n}rnk([s_{i}^{-1}]_{2})=k$. Show
that the product $s\cdot\widetilde{a}\cdot s^{-1}$ belongs to
$e\wr\widetilde{A_{n}}$. Denote representation by tableaux of
$s\cdot\widetilde{a}\cdot s^{-1}$ by $[b_{1}]_{1},\
[b_{1},b_{2},\ldots,b_{n}]_{2}$. Since of definition of
$e\wr\widetilde{A_{n}}$ we know that an element $\widetilde{a}\in
e\wr\widetilde{A_{n}}$ satisfy the following property of parity
$\sum_{i=1}^{n}rnk([a_{i}]_{2})=2k,k\in\mathbb{N}.$ Then taking into account
multiplication rule in a wreath product of groups and a property (4) we have
$[b_{1}]_{1}=[s_{1}\cdot e\cdot s_{1}^{-1}]_{1}=[e]_{1}$, and also
$\sum_{i=1}^{n}rnk([b_{i}]_{2})\\!=\\!\sum_{i=1}^{n}rnk([s_{i}]_{2})+\sum_{i=1}^{n}rnk([a_{i}]_{2})-2a+\sum_{i=1}^{n}rnk([s_{i}^{-1}]_{2})-2b=k+2m-2a+k-2b=2(k+m-a-b)$.
Hence the sum is even number this satisfy the condition of belonging to
$e\wr\widetilde{A_{n}}$. Thus, $s\cdot\widetilde{a}\cdot s^{-1}\in
e\wr\widetilde{A_{n}}$. That accomplished the proof. ∎

Let us denote the element of $e\wr\widetilde{A_{n}}$ having tableaux
presentation $[e]_{1},\ [(12),(12),e,e\ldots,e]_{2}$ by $\widetilde{a}$.

######  Proposition 2.

The normal closure $N_{S_{n}\wr S_{n}}(\widetilde{a})$ of $\widetilde{a}$ is
$e\wr\widetilde{A_{n}}$.

###### Proof.

To prove it, it is sufficient to show that $N(\widetilde{a})$ is closed in
$e\wr\widetilde{A_{n}}$ with respect to multiplication and conjugation.

1) this element of $N(\widetilde{a})$ is closed with respect of conjugation by
elements of $S_{n}\wr S_{n}$ is the subgroup $N(\widetilde{a})$ i.e. $\forall
s\in S_{n}\wr S_{n}:s\cdot a\cdot s^{-1}\in N(\widetilde{a})$.

2) multiplication of two elements $\forall a,b\in N(\widetilde{a}):$ satisfies
the closeness property $a\cdot b\in N(\widetilde{a})$.
Denote an element of $e\wr\widetilde{A_{n}}$ having form of the tableaux
$[e]_{1},\ [\pi_{1},\pi_{2},\ldots,\pi_{n}]_{2},$ where
$\pi_{1},\pi_{2},\ldots,\pi_{n}\in S_{n}$ and
$\sum_{i=0}^{n}rnk([\pi_{i}]_{2})=2k,k\in\mathbb{N}$ by
$a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}$. In particular
$\widetilde{a}=a_{(12),(12),e,\ldots,e}$. We make the constructive prove
showing that every element satisfying condition 7 belongs to normal clossuare
of element $\widetilde{a}$ i.e. belongs to $N(\widetilde{a})$.

(a) $\pi_{1}=(i_{1}i_{2}),\pi_{2}=(j_{1}j_{2}),\pi_{3}=\ldots=\pi_{n}=e$. Then
$a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}=s\cdot\widetilde{a}\cdot s^{-1}\in
N(\widetilde{a}),$ where $s$ has tableaux of form: $[e]_{1},\
[((1i_{1})(2i_{2})),((1j_{1})(2j_{2})),e,e\ldots,e]_{2}$.

(b) $\pi_{1}=(i_{1}i_{2})(i_{3}i_{4}),\,\pi_{2}=\ldots=\pi_{n}=e$. Then an
element $a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}$ can be expressed as product of
elements of type (a) where $\pi_{2}=(1,2)$:
$a_{(i_{1}i_{2}),(1,2),\ldots,\pi_{n}}\cdot
a_{(i_{3}i_{4}),(1,2),\ldots,\pi_{n}}\in N(\widetilde{a}).$ Since $A_{n}$ can
be generated by pairs of transpositions then $A_{n}\subseteqq
N(\widetilde{a})$.

Analogous case is $\pi_{2}=(j_{1}j_{2})(j_{3}j_{4}),$
$\pi_{1}=\pi_{3}=\ldots=\pi_{n}=e$ and the similar reasoning as above are
applicable.

(c) If $\pi_{1}=(i_{1}i_{2})\ldots(i_{2x-1}i_{2x}),$
$\pi_{2}=(j_{1}j_{2})\ldots(j_{2y-1}j_{2y}),\pi_{3}=\ldots=\pi_{n}=e$. Without
losing the generality, we assume $x\geqslant y$. Since
$rnk(\pi_{1}+\pi_{2})=2k$, then $(x-y)\equiv 0(mod2)$, since:
$a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}=a_{(i_{1}i_{2}),(j_{1}j_{2}),e\ldots,e}\cdot\ldots\cdot
a_{(i_{2x-1}i_{2x}),(j_{2x-1}j_{2x}),e\ldots,e}\cdot
a_{(i_{2x+1}i_{2x+2})(j_{2x+3}j_{2x+4}),e\ldots,e}\cdot\ldots$ $\cdot
a_{(i_{2y-4}i_{2y-3})(j_{2y-1}j_{2y}),e\ldots,e}\in N(\widetilde{a})$. Hence
such elements are the product of the elements which are described above in the
items (a) and (b).

(d) If $\pi_{1}=(i_{1}i_{2})\ldots(i_{4x-1}i_{4x}),$
$\pi_{2}=\ldots=\pi_{n}=e$. Then $a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}$ we can
write: $a_{(i_{1}i_{2})(i_{3}i_{4}),e...,e}\cdot\ldots\cdot
a_{(i_{4x-4}i_{4x-3})(j_{4x-1}j_{4x}),e\ldots,e}\in N(\widetilde{a}).$ Since
such elements are product of elements from item (b).

(e) If $\pi_{i}\neq e,\pi_{j}\neq e,rnk(\pi_{i}+\pi_{j})=2z,$
$z\in\mathbb{N},\pi_{k}=e,k\neq i,j$. Then
$a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}=\\\ =s\cdot
a_{\pi_{1},\pi_{2},e,\ldots,e}\cdot s^{-1}\in N(\widetilde{a}),$ where $s$ has
tableau presentation: $[(1i)(2j)]_{1},\ [e\ldots,e]_{2}$. So such elements are
conjugated to the elements described in (c).

(f) If $\pi_{i}\neq e,rnk(\pi_{i})=2z,z\in\mathbb{N},$ $\pi_{k}=e,k\neq i$.
Then $a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}=s\cdot a_{\pi_{1},e,\ldots,e}\cdot
s^{-1}\in N(\widetilde{a}),$ where $s$ has presentation: $[(1i)]_{1},\
[e\ldots,e]_{2}$. So such elements are conjugated to the elements described in
(d).

(g) If $\pi_{i}\neq e,i=\overline{1,n},$ and the order of permutation is even
i.e. $\sum_{i=1}^{n}rnk([\pi_{i}])=2z,z\in\mathbb{N}$ then
$a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}$ is a product of the elements described in
(e), (f).

So the group closure is $e\wr\widetilde{A_{n}}$. ∎

######  Remark 3.

The previous statement holds for the normal closure $N(\widetilde{b})$ of
$\widetilde{b}=a_{e,\ldots,e,(i_{1}i_{2}),e,\ldots,e,(j_{1}j_{2}),e,\ldots,e},\
i_{1},i_{2},j_{1},j_{2}\in\\{1,2,\ldots,n\\}$.

###### Proof.

To established the proof it is sufficient to show $\widetilde{a}\in
N(\widetilde{b})$, since inverse inclusion follows from previous Statement.
$a_{e,\ldots,e,(12),e,\ldots,e,(12),e,\ldots,e}=a_{e,\ldots,e,(i_{1}1i_{2}2),e,\ldots,e,(j_{1}{1}2j_{2}),e,\ldots,e}\cdot$
$\cdot\widetilde{b}\cdot(a_{e,\ldots,e,(i_{1}1i_{2}2),e,\ldots,e,(j_{1}1j_{2}2),e,\ldots,e})^{-1}\in
N(\widetilde{b}).$
Suppose an expressed element has non trivial permutations on coordinates $i$
and $j$. Therefore

$\widetilde{a}=s\cdot a_{e,\ldots,e,(12),e,\ldots,e,(12),e,\ldots,e}\cdot
s^{-1}\in N(\widetilde{b}),$

where $s$ has tableau form as $[(1i)(2j)]_{1},\ [e\ldots,e]_{2}$. ∎

Let us denote by $\widetilde{c}$ an element of $e\wr\widetilde{A_{n}}$ having
the tableaux presentation $[e]_{1},\ [(123),e,e,e\ldots,e]_{2}$.

######  Proposition 4.

The normal closure $N(\widetilde{c})$ of $\widetilde{c}$ is
$e\wr\widetilde{A_{n}}$.

###### Proof.

We use for the proving notifications of previous statements.

(a) If $\pi_{1}\in A_{n},\pi_{2}=\ldots=\pi_{n}=e$ then the element
$\widetilde{c}$ has the form $c_{\pi_{1},e,\ldots,e}$.
Therefore the element $a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}\in N(\widetilde{c})$
can be obtained by using of transformation of first and second type from
Proposition 2 (conjugations and multiplications) of an element
$\widetilde{c}$.

Since $A_{n}\lhd S_{n}$ and $A_{n}$ is prime, then a normal closure its
elements coincides with hole $A_{n}$. Hence an element
$a_{\pi_{1},\pi_{2},\ldots,\pi_{n}}\in N(\widetilde{c})$ where $\pi_{i}$ is
3-cycle, can be expressed from $\widetilde{c}$ by using transformations from
prof of previous Statements.

(b) If $\pi_{i}\neq e,\pi_{i}\in A_{n}$, but for another $k\neq i$ we set
$\pi_{k}=e,$ then

$a_{e,\ldots,\pi_{i},\ldots,e}=s\cdot a_{\pi_{1},e,\ldots,e}\cdot s^{-1}\in
N(\widetilde{c}),$

where $s$ is presented by the tableau: $[(1i)]_{1},\ [e\ldots,e]_{2}$.
Therefore, $a_{e,\ldots,\pi_{i},\ldots,e}$ are conjugated with elements from
item (a).

(c) If $\pi_{i}\in A_{n},$ for every $i=\overline{1,n}$. Then
$c_{\pi_{1},\pi_{2},\ldots,\pi_{n}}$ can be expressed as a product of elements
described in the item (b).

Thus, all elements of $e\wr\widetilde{A_{n}}$ are constructed in the items
(a)-(c), that is what accomplishes the proof. ∎

######  Corollary 5.

Previous Statement remains to be correct for the normal closure
$N(\widetilde{b})$ of
$\widetilde{b}=a_{e,\ldots,e,(i_{1}i_{2}i_{3}),e,\ldots,e},\
i_{1},i_{2},i_{3}\in\\{1,2,\ldots,n\\}$.

###### Proof.

To prove the statement it is sufficient to establish $\widetilde{a}\in
N(\widetilde{b})$, because of the belonging $\widetilde{b}\in
N(\widetilde{a})$ follows from the $\widetilde{a}\in N(\widetilde{b})$.

Let us prove it. So we have
$a_{e,\ldots,e,(123),e,\ldots,e}=$
$a_{e,\ldots,e,(i_{1}1i_{2}2i_{3}3),e,\ldots,e}\cdot\widetilde{b}\cdot(a_{e,\ldots,e,(i_{1}1i_{2}2i_{3}3),e,\ldots,e})^{-1}\in$
$N(\widetilde{b}).$

Without losing the generality we can set non trivial permutations in obtained
an element is on coordinate $j$. Therefore using the conjugation by the
tableau $[(1j)]_{1},\ [e,\ldots,e]_{2}$ we obtain the mentioned above element
$\widetilde{a}$:

$\widetilde{a}=s\cdot a_{e,\ldots,e,(123),e,\ldots,e}\cdot s^{-1}\in
N(\widetilde{b}).$

Thus, the required element $\widetilde{a}$ was expressed in
$N(\widetilde{b})$. ∎

######  Definition 8.

An element of iterated wreath product
$\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$ is called finite if, starting
from some level, all vertex permutations of the level are trivial.

A mapping $g|_{v}$ is automorphism of tree $X^{*}$ and $g|_{v}$ calls
_restriction_ of $g$ on word $v$ or the _state_ $g$ in word $v$. This mapping
$g|_{v}$ defined by the following rule $g|_{v}:X^{*}\rightarrow X^{*}$ iff
$g(vx)=uy$, where $|x|=|y|$ and $g(vx)=uy$.

Automorphism $g$ calls _finite_ if it exists $n\geq 1$ such that $g|_{v}=e$
for all ${v}\in X^{*}$, $|x|\geq n$. The set of all finite automorphisms
generate the group $Aut_{f}X^{*}$.

If $H$ is proper normal subgroup of depth $k$ and
$H\triangleleft\wr^{m}_{i=1}S_{n_{i}}$, where $k\in N$, $n_{i}\geq 5$ $m$ can
tends to $\infty$, then subgroup $H$ contains finite elements which have even
permutations in all vertexes of $k$-th level states. Such permutation can be
generated as left based commutators of level permutation of
$\wr^{m}_{i=1}S_{n_{i}}$ with depth no less than $k$. Now we are going to
prove it.

Let $H$ be a proper normal subgroup with depth $k$.

######  Corollary 6.

Subgroup $H$ contains all finite elements with even level permutations.

Let $v$ be vertex of $k$-th level of $AutX^{m}$ with non-trivial vertex
permutation $\sigma$, without loosing the generality we set that
$\sigma=(1,2,...,l),\,\sigma\in S_{n_{k}}$. In general case the permutation
$\sigma$ can consists of some cycles but for proof it is enough to consider
one cycle. Therefore state of automorphism in $v$ can be presented in the form
of the following tableau
$x=(b_{1},b_{2},...,b_{l-1},b_{l})\sigma,\,b_{i}\in\wr^{m-1}_{i=1}S_{n_{i}},\,n_{k}\leq
m,\,l=n_{k}$. Let $y=(e,e,...,e,e)\sigma_{1}$, where
$\sigma_{1}=(1,2,...,l-2)$, then
$z=yxy^{-1}=(b_{2},b_{3},...,b_{l-2},b_{1},b_{l-1},b_{l})\sigma,$ and
$zx^{-1}=yxy^{-1}x^{-1}=(a_{1},...,a_{l-2},e,e)\sigma$, wher
$a_{1}=b_{2}b_{1}^{-1}$, because of
$x^{-1}=(b_{l}^{-1},b_{1}^{-1},b_{2}^{-1},...,b_{l-1}^{-1})\sigma^{-1}$. Now
we see that $zx^{-1}$ is even permutation, moreover $zx^{-1}=[y,x]$.

Consider the element of $H$ stabilizing elements in vertexes $1,...,l-2$ i.e.
the element of form $g=(e,e,e,e,e)(l-1,l)$. Such $g$ can be chosen from
$G\backslash H$ with respect to the restriction of action in $v$ is such that
$g|_{v}=(,e,e,e,e,e)(l-1,l)$. Due to $H$ is normal in $G$ the inclusion
$gyxy^{-1}x^{-1}g^{-1}\in H$ holds. We chose $x\in St_{G}(v_{1})\bigcap
St_{G}(v_{2})\bigcap...\bigcap St_{G}(v_{l-2})$ and $g$ from complementary
vertex stabilizer intersection of $y$ i.e. $g\in St_{G}(v_{l-1})\bigcap
St_{G}(v_{l})$.

Now we conjugate $yxy^{-1}x^{-1}$ by such $g=(e,e,e,e,e)(l-1,l)$, which is
permutable only on trivial elements in tuple $(a_{1},...,a_{l-2},e,e)\sigma$
fixing all non trivial elements, this tuple presents the element
$yxy^{-1}x^{-1}$ in the form of wreath recursion.

The obtained element $gyxy^{-1}x^{-1}g^{-1}=(a_{1},a_{2},...,a_{l-2},e,e)\pi$
we multiply by the inverse element to commutator $[y,x]$:
$[x,y]gyxy^{-1}x^{-1}g^{-1}=(zx^{-1})^{-1}gxyx^{-1}y^{-1}g^{-1}=(zx^{-1})^{-1}gzx^{-1}g^{-1}=[x,y]g[y,x]g^{-1}=(e,e,e,e,e)(\tau)$,
where $\tau=[\sigma_{1},\sigma]\pi$, $\sigma_{1}$ is permutation from the
first level of the state of portrait automorphism $y$, $\sigma$ is a root
permutation of state of automorphism of $x$ in $v$ and $\pi$ is root
permutation of state of $gyxy^{-1}x^{-1}g^{-1}$ in $v$. In terms of
commutators $(zx^{-1})^{-1}gzx^{-1}g^{-1}$ can be presented as
$[xz^{-1},g]=[[y,x],g]$. Thus, we obtain the element $[[y,x],g]$ having
trivial states in vertexes of $k+j$ levels and a permutation $\rho$ on $k$-th
level in $v$. Being a commutator $\rho$ is even permutation from $S_{n_{k}}$
therefore $\rho\in A_{n_{k}}$.

In the similar way elements of arbitrary depth $i$ having an even permutation
on $j$-th level and trivial permutations on other levels can be constructed.

If we consider $S_{3}\wr S_{2}$ (active subgroup is the left) then we find 7
proper normal subgroups between 9 normal subgroups:
$[["1",1],["A_{4}",1],["C_{2}",1],["C_{2}\times A_{4}",1],["C_{2}\times
C_{2}",1],["C_{2}\times C_{2}\times C_{2}",1],["C_{2}\times
S_{4}",1],["S_{4}",2]].$

######  Remark 7.

The following isomorphism $S_{3}\wr S_{2}\simeq C_{2}\times S_{4}$ is true.

###### Proof.

The proof follows from direct computations. ∎

For instance we analysis all normal subgroups of $S_{3}\wr S_{3}$. Computation
in GAP lead us to following results and conclusions:

$S_{3}\wr S_{3}$. The group $S_{3}\wr S_{3}$ has 10 normal subgroups, where
$G_{0}=E$ and $G_{1}=S_{3}\wr
S_{3}=\langle(1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8),(1,4,7)(2,5,8)(3,6,9),(1,4)(2,5)(3,6)\rangle$.
The list of proper normal subgroups is the following:
$G_{2}=\langle(7,8,9),(4,5,6),(1,2,3)\rangle$,
$G_{3}=\langle(7,8,9),(4,5,6),(1,2,3),(2,3)(8,9),(2,3)(5,6)\rangle$,
$G_{4}=\langle(7,8,9),(4,5,6),(1,2,3),(1,4,7)(2,5,8)(3,6,9),(2,3)(5,6)\rangle$,
$G_{5}=\left\langle((7,8,9),(4,5,6),(1,2,3),(1,4)(2,6)(3,5)(8,9),(1,4,7)(2,5,8)(3,6,9),\right.\\\
\left.(2,3)(5,6)\right\rangle$,
$G_{6}=\langle(7,8,9),(4,5,6),(1,2,3),(1,4)(2,5)(3,6),(1,4,7)(2,5,8)(3,6,9),(2,3)(5,6)\rangle$,
$G_{7}=\langle(2,3)(5,6)(8,9),(7,8,9),(4,5,6),(1,2,3)\rangle$,
$G_{8}=\langle(8,9),(5,6),(2,3),(7,8,9),(4,5,6),(1,2,3)\rangle$,
$G_{9}=\langle(1,4,7)(2,5,8)(3,6,9),(8,9),(5,6),(2,3),(7,8,9),(4,5,6),(1,2,3)\rangle$.

In total we have 8 proper normal subgroups in $S_{3}\wr S_{3}$.

By analyzing their structures in terms of a semidirect product and
determination them as the subgroup of wreath product with respect of their
generators we obtain:

1) $S_{3}\wr\widetilde{A_{3}}\simeq(((C_{3}\times C_{3}\times
C_{3})\rtimes(C_{2}\times C_{2}))\rtimes C_{3})\rtimes C_{2},3]$ here are 3
non equal isomorphic subgroups, also we compute
$ord(S_{3}\wr\tilde{A_{3}})=((6^{3}):2)\cdot 6=3^{4}\cdot 2^{3}=108\cdot
6=648$.

2)
$\langle(1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8),(1,4,7)(2,5,8)(3,6,9),(1,4)(2,5)(3,6)\rangle=S_{3}\wr
S_{3}$.

All subgroups, except one having the number 7, from this list can be easily
identified. We consider the following definition related to $G_{7}$.

######  Definition 9.

The set of elements from $S_{n}\wr S_{n},n\geqslant 5$ of the form: $[e]_{1},\
[a_{1},a_{2},\ldots,a_{n}]_{2},$ satisfying the following condition

$\sum_{i=1}^{n}rnk([a_{i}]_{2})=2k,k\in\mathbb{N},$ (5)

we will call set of type $\widetilde{A}^{(0)}$.

######  Proposition 8.

The set of elements of type $\widetilde{A}^{(0)}$ forms the subgroup
$E\wr\widetilde{A_{n}}\lhd S_{n}\wr S_{n}$

######  Lemma 9.

The normal closure ${{N}_{{{S}_{n}}\wr{{S}_{n}}}}(E\wr\widetilde{A_{n}})$ is
$E\wr\widetilde{{A}_{n}}$.

###### Proof.

According to Definition 9 the product of elements of passive subgroup of
wreath product satisfy condition of parity.

It is easy to check that the normal closure of the subgroup
$E\wr\widetilde{A_{n}}$ in ${{S}_{n}}\wr{{S}_{n}}$ is itself
$E\wr\widetilde{A_{n}}$.

Indeed, conjugation by arbitraty $g\in{{S}_{n}}\wr{{S}_{n}}$ an arbitrary
permuted element does not change the parity of the element $h\in
E\wr\widetilde{A_{n}}$.

Also, elements of this type ${\widetilde{A}^{(0)}}$ described in Definition 9
with an even product of all substitutions of base group satisfy the condition
of belonging to the commutator subgroup $B^{\prime}$ of wreath product passive
group $B$ described in the author’s article and in Meldrum’s book [11, 13]. ∎

######  Definition 10.

A subgroup in $S_{n}\wr S_{n}$, we call a subgroup of type
$\widetilde{T_{n}}$, if it contains elements of the form:
1\. first type are elements of $\widetilde{T_{n}}$ having the form $e\wr
A_{n}\subset\widetilde{T_{n}}$,
2\. second type of $\widetilde{T_{n}}$ consists of elements with the tableau
presentation:
$[e]_{1},\ [\pi_{1}\ldots,\pi_{n}]_{2}$, that $\forall i=1,\ldots
n:rnk(\pi_{i})=2k+1,k\in{\textbf{N}}$ or equivalently $\pi_{i}\in
S_{n}\setminus A_{n}$ (such elements be called elements of second type from
subgroup $\widetilde{T_{n}}$).

One easy can validates a correctness of this definition i.e. that this
subgroup is the group as well as its normality.

We denote $S_{3}\wr S_{3}$ by $H_{1}$ and we compute that
$[H_{1}=((((C_{3}\times C_{3}\times C_{3})\rtimes(C_{2}\times C_{2}))\rtimes
C_{3})\rtimes C_{2})\rtimes C_{2},1].$ Order of $H_{1}$ is 1296.
We find the structures of these 8 proper normal subgroups of $S_{3}\wr S_{3}$:
$H_{2}=[(((C_{3}\times C_{3}\times C_{3})\rtimes(C_{2}\times C_{2}))\rtimes
C_{3})\rtimes C_{2},3],$
$H_{3}=[((C_{3}\times C_{3}\times C_{3})\rtimes(C_{2}\times C_{2}))\rtimes
C_{3},1],$
$H_{4}=[(C_{3}\times C_{3}\times C_{3})\rtimes(C_{2}\times C_{2}),1],$
$H_{5}=[(C_{3}\times C_{3}\times C_{3})\rtimes C_{2},1],$
$H_{6}=[C_{3}\times C_{3}\times C_{3},1],$
$H_{7}=[S_{3}\times S_{3}\times S_{3},1]].$

Thus, $S_{3}\wr S_{3}$ contains 8 properly normal subgroups (one of them,
numbered by 2, determined by the same products but which has non isomorphic
three different structures). Furthermore form of their generators we can
deduce the splitability of this groups.

Totally we have 8 proper normal subgroups in $S_{3}\wr S_{3}$.

By analyzing their structures in terms of a semidirect product and
determination them as the subgroup of wreath product with respect of their
generators we obtain:

1) Consider the three isomorphic subgroups having GAP structure description
$H_{2}$ = ["(((C3 x C3 x C3) : (C2 x C2)) : C3) : C2", 3 ] here are 3 non
equal isomorphic subgroups of order 648. Algebraic structure of them is the
following:

$H_{2}\simeq\left(\left(\left(C3\text{ }\times\text{ }C3\text{ }\times\text{
}C3\right)\rtimes\left(C2\text{ }\times\text{ }C2\right)\right)\rtimes\text{
}C3\right)\text{ }\rtimes\text{ }C2,$

these subgroups have order 648 and identifiers according to GAP system of this
small group: id =703, 704 and 705 respectively.

At the same time, we compute $ord(S_{3}\wr\widetilde{A_{3}})=((6^{3}):2)\cdot
6=3^{4}\cdot 2^{3}=108\cdot 6=648$.

We will denote it by ${{H}_{703}},\,\,\,{{H}_{704}}$ and ${{H}_{705}}$
correspondingly to their "id" in GAP. To classify these groups we compute its
commutator subgroups and centers. The subgroup $H{{{}^{\prime}}_{703}}$ and
$H{{{}^{\prime}}_{704}}$ have order 324 and GAP idetifier is 160, this order
is exactly half of 648. This means that commutator of these subgroups is the
same therefore ${{H}_{703}}$ and ${{H}_{704}}$ are isomorphic but they are
embedded in ${{S}_{3}}\wr{{S}_{3}}$ by in different ways as different copies.
The center of this subgroups are trivial subgroup. The commutator subgroup of

${{S}_{3}}\wr{{\widetilde{A}}_{3}}$

is exactly ${{A}_{3}}\wr{{\widetilde{A}}_{3}}$, so active subgroup has order
in 2 times less then order of ${{S}_{3}}$. Thus, we can conclude that
${{H}_{703}}\,\simeq\,\,{{S}_{3}}\wr{{\widetilde{A}}_{3}}$ as well as
${{H}_{704}}\,\simeq\,\,{{S}_{3}}\wr{{\tilde{A}}_{3}}$. The center of
${{S}_{3}}\wr{{\tilde{A}}_{3}}$ is the trivial subgroup too that confirms
isomorphism $H_{703}\simeq H_{704}\simeq{{S}_{3}}\wr{{\widetilde{A}}_{3}}$.

In $H_{705}$ the homomorphism from top group of semidirect product in
$Aut{{\widetilde{A}}_{3}}$ has another structure of the kernel,
$ord(H^{\prime}_{705})=108$ which in 3 time less than order of
$H^{\prime}_{703}$.

On the other side the normal subgroup $A_{3}\wr S_{3}$ has the commutator
subgroup $e\wr\widetilde{A}_{3}$ with the same order 108. Thus, the third copy
of $H_{2}$ from the GAP list is

$A_{3}\wr S_{3}=H_{705}.$

2)
$\langle(1,2,3),(1,2),(4,5,6),(4,5),(7,8,9),(7,8),(1,4,7)(2,5,8)(3,6,9),(1,4)(2,5)(3,6)\rangle=S_{3}\wr
S_{3}$.

3) $T_{3}=(C_{3}\times C_{3}\times C_{3})\rtimes C_{2}$ taking into account
that $C_{3}\simeq A_{3}$ we see that in general case when $G=S_{n}\wr S_{n}$
this subgroup has structure
$\tilde{T_{n}}\simeq(\mathop{\underbrace{A_{n}\times A_{n}\times\cdots\times
A_{n}}}\limits_{n})\rtimes C_{2}$.

4) $((C_{3}\times C_{3}\times C_{3})\rtimes(C_{2}\times C_{2}))\rtimes
C_{3}=A_{3}\wr\tilde{A_{3}}$

######  Remark 10.

If $G=S_{n}\wr S_{m}$ then the subgroup of type $\widetilde{T_{m}}$ has
structure ${T_{n,m}}\simeq(\mathop{\underbrace{A_{m}\times
A_{m}\times\cdots\times A_{m}}}\limits_{n})\rtimes C_{2}$.

The parity of product of permutations tuple $(g_{1},g_{2},\cdots,g_{n})$,
where $g_{i}\in A_{m}$, on $\delta\in C_{2}$ depends on homomorphic image of
$\delta\in C_{2}$ in $Aut(\mathop{\prod}\limits^{n}_{i=1}A_{m}^{(i)})$ as an
element $((g_{1},g_{2},\cdots,g_{n}),\delta)$ of the semidirect product
$\mathop{\prod}\limits^{n}_{i=1}A_{m}^{(i)}\rtimes C_{2}$.

We recall that a center of the group $(\mathcal{A},X)\wr\mathcal{B}$ is direct
product of normal closure of center of diagonal of $Z(\mathcal{B}^{n})$ i.e.
($E\times Z(\bigtriangleup(\mathcal{B}^{n}))$), trivial an element, and
intersection of $(\mathcal{K})\times E$ with $Z(\mathcal{A})$ [11].

Note that ${\widetilde{T}_{m}}$ is generalization of diagonal subgroup. Taking
into account formulated above properties of ${{S}_{n}}\wr{{S}_{m}}$ we
conclude that $E\times\Delta Z(S_{m}^{n})$ [11] is trivial, where $Z(\Delta
S_{m}^{n})$ denotes the diagonal subgroup of
$S_{m}^{n}=\prod\limits^{n}_{1}S_{m}$. But the following generalization of the
diagonal of base of wreath product is a non-trivial normal subgroup. A passive
group of $N\triangleleft A\wr B$ consists of equivalence classes of an
algebraic subgroup $B$, where conjugation by elements $A\wr B$ of acts
imprimitive. These generalization of diagonal subgroups of base of wreath
product are ${\widetilde{{T}_{m}}},\,{T}_{n,m}$ and
$E\wr{\widetilde{{A}_{m}}}$.

In this generalization a passive subgroup of $N\triangleleft A\wr B$ consists
of equivalence classes of an algebraic subgroup $B$, where conjugation by
elements $A\wr B$ of acts imprimitive. These generalized diagonal subgroups
are ${\widetilde{T}_{m}}$ and $E\wr\widetilde{{A}_{m}}$.

The intersection of kernel of action of active subgroup and its center is
trivial because center of ${{S}_{n}}$ is trivial and kernel of action
$({{S}_{n}},\,{{N}_{n}})$, where ${{N}_{n}}=\\{1,...,n\\}$ is trivial because
of action is faithful. Also in this case of the center of diagonal of base of
wreath product ${{S}_{n}}\wr{{S}_{m}}$ i.e $\Delta Z(S_{m}^{n})$ [11] is
trivial too. Therefore usual diagonal subgroup of base subgroup
${{{{S}_{m}}}^{n}}$ from ${{S}_{n}}\wr{{S}_{m}}$ does not form normal
subgroup. Thus, center of ${{S}_{n}}\wr{{S}_{m}}$ is trivial.

The cycle-type of a permutation of $S_{n}$ is the partition of $n$ obtained
from the lengths of the cycles that appear in its decomposition into product
of disjoint cycles.

In this paper we will denote the cycle-type of a permutation $\omega$ by
$ct(\omega)$.

We recall that group is _monolithic_ if the intersection of all nontrivial
normal subgroups is nontrivial; the _monolith_ is this intersection.

######  Theorem 11.

Proper normal subgroups in $S_{n}\wr S_{m},\,n,\,m\geqslant 3,\,n,\,m\neq 4$
are the following 2 types:

1\. subgroups, acting only on second level (stabilizing the first level),
which are the following: $e\wr A_{m}$, $e\wr\widetilde{A_{m}}$,
$\widetilde{T_{m}}$, $e\wr S_{m}$;

2\. subgroups, acting on both levels, which are the following:
$S_{n}\wr\widetilde{A_{m}}$ and another subgroup determined by the same
semidirect product as $H_{2}$, $A_{n}\wr\widetilde{A_{m}}$, $A_{n}\wr S_{m}$.

Thus, total number of proper normal subgroups in $S_{n}\wr
S_{m},\,n,m\geqslant 3,n,m\neq 4$ is 8.

All of these groups are splittable groups. The _monolith_ of $S_{n}\wr S_{m}$
is $e\wr\widetilde{A_{m}}$ if $m\equiv 0(mod2)$.

The subgroup $S_{n}\wr\widetilde{A_{m}}$ appears only in $W=S_{n}\wr S_{m}$.

Proper normal subgroups in $S_{n}\wr S_{m},\,n,\,m\geqslant 3,\,n,\,m\neq 4$
are the following 2 types:

Thus, total number of proper normal subgroups in $S_{n}\wr
S_{m},\,n,m\geqslant 3,n,m\neq 4$ is 8.

Note that subgroup $A_{n}\wr\widetilde{A_{m}}=W^{\prime}$. Also
$S_{n}\wr\widetilde{S_{m}}$, and $A_{n}\wr S_{m}$ is the new normal proper
subgroup of $S_{n}\wr\widetilde{S_{m}}$ relatively to the normal subgroups of
$A_{n}\wr S_{m}$ which has 5 proper subgroups.

###### Proof.

We recall that in case of trivial action $G$ on $X$ a wreath product $(G,X)\wr
H$ degrades in $G\times H^{X}$. Since if $G=E$ we have trivial action then on
$X$ then subgroup $E\times S_{m}^{n}$ appears.

First of all, note that all our normal subgroups contain subgroup
$e\wr\widetilde{A_{m}}$. If we prove the normality of $e\wr\widetilde{A_{m}}$
then last two facts would implies that $S_{n}\wr\widetilde{S_{m}}$ is
_monolithic_ group and its _monolith_ is $e\wr\widetilde{A_{m}}$. To prove the
normality of $e\wr\widetilde{A_{m}}$ as well as $A_{n}\wr\widetilde{S_{m}}$
and $S_{n}\wr\widetilde{S_{m}}$ it is sufficient to validate the embedednes
(incorporation) of the normal closure of its generators to the correspondent
original subgroup $e\wr\widetilde{A_{m}}$, $A_{n}\wr\widetilde{S_{m}}$ and
$S_{n}\wr\widetilde{S_{m}}$ for $n,m\geqslant 3,n,m\neq 4$.

Generators of each of subgroups will be written in a similar way with the
generators of normal subgroups $S_{3}\wr S_{3}$. The proof of split ups of any
these subgroups will follow from the form of their generators elements.

1\. a) $e\wr A_{m}$: Its generating elements have the following form:
$a_{e,\ldots,e,(i_{1}i_{2}i_{3}),e,\ldots,e},$ where
$i_{1},i_{2},i_{3}\in\\{1,2,\ldots,m\\}$, since $A_{m}$ and $[e]_{1},\
[(12),(12),e,e\ldots,e]_{2}$ which denoted by $\widetilde{a}$. The normal
closure of $\widetilde{a}$ is found in 2. Its coincide with $e\wr A_{m}$.
Moreover, $A_{n}$ can be generated by all 3-cycles and all 3-cycles can be
generated by conjugating $(i_{1}i_{2}i_{3})$ by even substitutions, such
substitutions are in $S_{m}$. Proposition 4 and Corollary 5 show us that the
conjugated generator $a_{e,\ldots,e,(i_{1}i_{2}i_{3}),e,\ldots,e}$ by elements
of $S_{n}\wr S_{m}$ pertains to $e\wr A_{m}$ too.

_Normality of the subgroup_. Consider an arbitrary generator $a$, without
losing the generality $a$ can be in of the form:
$a=a_{e,\ldots,e,(i_{1}i_{2}i_{3}),e,\ldots,e},$
$i_{1},i_{2},i_{3}\in\\{1,2,\ldots,m\\}\in A_{m}$. Then for any element $b\in
S_{n}\wr S_{m}$, that has tableau presentation in the form:
$[\pi_{0}]_{1},[\pi_{1},\ldots,\pi_{n}]_{2},\pi_{0}\in
S_{n},\pi_{1},\ldots,\pi_{n}\in S_{m}$ we have: $[b\cdot a\cdot
b^{-1}]_{1}=[\pi_{0}\cdot e\cdot\pi_{0}^{-1}]=[e]_{1};$ $[b\cdot a\cdot
b^{-1}]_{2}=[b(x)\cdot a(x^{\pi_{0}})\cdot
b^{-1}(x^{\pi_{0}\cdot\pi_{0}^{-1}})]_{2}=$
$=[\pi_{1}\cdot
e\cdot\pi_{1}^{-1},\ldots,\pi(x^{\pi_{0}})\cdot(i_{1}i_{2}i_{3})\cdot\pi(x^{\pi_{0}})^{-1},\ldots,\pi_{n}\cdot
e\cdot\pi_{n}^{-1}]_{2}=$
$=[e,\ldots,e,\pi(x^{\pi_{0}})\cdot(i_{1}i_{2}i_{3})\cdot\pi(x^{\pi_{0}})^{-1},e,\ldots,e]_{2}.$

Obviously, the mentioned above element pertain to $e\wr A_{m}$ because of
$\pi(x^{\pi_{0}})\cdot(i_{1}i_{2}i_{3})\cdot\pi(x^{\pi_{0}})^{-1}$ is an even
substitution.

b) _Generators of_ $e\wr\widetilde{A_{m}}$ are the following:
$a_{e,\ldots,e,(i_{1}i_{2}i_{3}),e,\ldots,e},$ where
$i_{1},i_{2},i_{3}\in\\{1,2,\ldots,m\\}$ and
$a_{(i_{1}i_{2}),e,\ldots,e,(i_{1}i_{2}),e,\ldots,e},\
i_{1},i_{2}\in\\{1,2,\ldots,m\\}$. The normal closure of these generators is
found in Proposition 2. This closure $N_{S_{n}\wr S_{n}}(\widetilde{a})$
equals to $e\wr\widetilde{A_{m}}$.

d) Consider $E\wr S_{m}$. Recall that two element generating set of $S_{m}$
has form $\tau=(1,2)$, $\lambda=(1,2,...,m)$. Without losing the generality we
can consider generators for $S_{3}$. Taking into consideration the mentioned
above generating set the generators of $e\wr S_{m}$ have the form:
$a_{e,\ldots,e,(i_{1}i_{2}i_{3}),e,\ldots,e},\ $ where
$i_{1},i_{2},i_{3}\in\\{1,2,\ldots,m\\}$ and
$a_{e,\ldots,e,(i_{1}i_{2}),e,\ldots,e},\
i_{1},i_{2},i_{3}\in\\{1,2,\ldots,m\\}$.

Let us verify the _normality of this subgroup_. It is obviously, because on
the first levels in result of the conjugation of generators by permutation of
$S_{n}$ always be identical substitution, and on the second level be any
substitution from $S_{m}$ which was required to prove in the theorem.

c) $\widetilde{T_{m}}$: Generators for elements of first type can be presented
by following way: $a_{e,\ldots,e,(i_{1}i_{2}i_{3}),e,\ldots,e},$ where
$i_{1},i_{2},i_{3}\in\\{1,2,\ldots,m\\}$ and generators for the second type:
$a_{(i_{1}i_{2}),\ldots,(i_{3}i_{4}),\ldots,(i_{5}i_{6})},i_{1},...,i_{6}\in\\{1,2,\ldots,m\\}$.

For generators of first type, the closedness in $\widetilde{T_{m}}$ with
respect to conjugation by elements of $S_{n}\wr S_{m}$ was shown in the item
a).

It remains to prove the same for the generators of second type. we consider
the generators of the form:
$a_{(i_{1}i_{2}),\ldots,(i_{1}i_{2}),\ldots,(i_{1}i_{2})},\in\widetilde{T_{m}}$.
Then for any element
$b\in S_{n}\wr S_{m}$, having tableau presentation
$[\pi_{0}]_{1},[\pi_{1},\ldots,\pi_{n}]_{2},\pi_{0}\in S_{n},$
$\pi_{1},\ldots,\pi_{n}\in S_{m}$ we have:

$\displaystyle[b\cdot a\cdot b^{-1}]_{1}=[\pi_{0}\cdot
e\cdot\pi_{0}^{-1}]=[e]_{1};$ (6) $\displaystyle[b\cdot a\cdot
b^{-1}]_{2}=[b(x)\cdot a(x^{\pi_{0}})\cdot
b^{-1}(x^{\pi_{0}\cdot\pi_{0}^{-1}})]_{2}=$
$\displaystyle=[\pi_{1}\cdot(i_{1}i_{2})\cdot\pi_{1}^{-1},\ldots,\pi_{1}\cdot(i_{1}i_{2})\cdot\pi_{n}^{-1}]_{2}=$
$\displaystyle=[\pi_{k}(x^{\pi_{0}})\cdot(i_{1}i_{2})\cdot\pi_{k}(x^{\pi_{0}})^{-1}]_{2}.$

It is obvious that the obtained element
$[\pi_{k}(x^{\pi_{0}})\cdot(i_{1}i_{2})\cdot\pi_{k}(x^{\pi_{0}})^{-1}]_{2}$ is
odd permutation, therefore $b\cdot a\cdot b^{-1}$ belongs to the second type.
This accomplished the prove for $\widetilde{T_{m}}$.

2\. Here we prove that the groups mentioned in item 2 are normal with using
the fact of normality in $S_{m}$ of subgroups appearing on second level. Such
groups are normal in $S_{m}$.

a) $A_{n}\wr\widetilde{A_{m}}$: Generators are the same as for
$e\wr\widetilde{A_{m}}$ that are
$a_{e,\ldots,e,(i_{1},i_{2},i_{3}),e,\ldots,e},$ and one new generator of
wreath product $A_{n}\wr\widetilde{A_{m}}$ with the following tableau
presentation:

$[(i_{1},i_{2},i_{3})]_{1},[e,\ldots,e]_{2},\
i_{1},i_{2},i_{3}\in\\{1,2,\ldots,n\\}.$

Taking into account that $A_{n}$ is generated by all $3$-cycles and the fact
that all 3-cycles can be generated by conjugating of $(i_{1},i_{2},i_{3})$ by
even substitutions $\pi$ we use for conjugation elements of form
$[\pi]_{1},[e,\ldots,e]_{2},\ \,i_{1},i_{2},i_{3}\in\\{1,2,\ldots,n\\}$. As a
result we see that normal closure of the new generators is $A_{n}\wr E$. And
normal closure of all generators is $A_{n}\wr\widetilde{A_{m}}$.

b) _Generating elements of_ $S_{n}\wr\widetilde{A_{m}}$ have the same
presentation as for $A_{n}\wr\widetilde{A_{m}}$ except one additional
generator presented by the next tableau:

$[(i_{1}i_{2})]_{1},[e,\ldots,e]_{2},\ i_{1},i_{2},\in\\{1,2,\ldots,n\\}.$

The last generator completes generation set of active group to generating set
of $S_{n}$, its normal closure pertain to $S_{n}\wr E$, the normal closure of
$(i_{1}i_{2})$ is considered above in item c) for generators of
$\widetilde{T_{m}}$. Therefore this completed generating set generates
$S_{n}\wr\widetilde{A_{m}}$.

c) _The generating set of_ $A_{n}\wr S_{m}$ are the same as for described
above $e\wr S_{m}$ except one additional generator in the normal closure which
has tableau presentation:

$[(i_{1}i_{2}i_{3})]_{1},[e,\ldots,e]_{2},\
i_{1},i_{2},i_{3}\in\\{1,2,\ldots,n\\}.$

As it is mentioned above $A_{n}$ is generated by all $3$-cycles and the fact
that all 3-cycles can be generated by conjugating of $(i_{1}i_{2}i_{3})$ by
even permutations. Therefore normal closure of additional generator
$[(i_{1}i_{2}i_{3})]_{1},[e,\ldots,e]_{2},\
i_{1},i_{2},i_{3}\in\\{1,2,\ldots,n\\}.$ ∎

The example of $S_{5}\wr S_{5}$ confirms last statement because of it gives us
8 proper normal subgroups too.

######  Remark 12.

We have to note that $A_{n}\wr A_{m}$ is not normal as in $S_{n}\wr S_{m}$ as
well as in $A_{n}\wr S_{m}$ because of wreath product having specific
conjugation as a result of action.

######  Theorem 13.

Proper normal subgroups in $A_{n}\wr S_{m},\,n,\,m\geqslant 3,\,n,\,m\neq 4$
are the following 3 types:

1\. subgroups, acting only on second level (stabilizing the first level),
which are the following: $E\wr\widetilde{A_{m}}$, $\widetilde{T_{m}}$, $E\wr
S_{m}$;

2\. subgroups, acting on both levels: $A_{n}\wr\widetilde{A_{m}}$

3\. subgroups with active vertexes only on first level: $A_{n}\wr E$.

Thus we have 5 in $A_{n}\wr{S_{m}}$ and 8 proper normal subgroups in $S_{n}\wr
S_{m}$ (active in the left).

###### Proof.

The proofs for all subgroups except one is similar to proofs in Theorem 11.
The proof of normality for $A_{n}\wr\widetilde{A_{m}}$ is such:

$A_{n}\wr\widetilde{A_{m}}$: Its generators are the same as for
$e\wr\widetilde{A_{m}}$ that are
$a_{e,\ldots,e,(i_{1}i_{2}i_{3}),e,\ldots,e},$ and one new generator of wreath
product $A_{n}\wr\widetilde{A_{m}}$ with the following tableau presentation:

$[(i_{1}i_{2}i_{3})]_{1},[e,\ldots,e]_{2},\
i_{1},i_{2},i_{3}\in\\{1,2,\ldots,n\\}.$

Taking in account that $A_{n}$ is generated by all $3$-cycles and the fact
that all 3-cycles can be generated by conjugating of $(i_{1}i_{2}i_{3})$ by
even substitutions $\pi$ we use for conjugation elements of form
$[\pi]_{1},[e,\ldots,e]_{2},\ \,\pi\in\\{1,2,\ldots,n\\}$. As a result we see
that normal closure of the last kind of generators is $A_{n}\wr E$. And normal
closure of all generators is $A_{n}\wr\widetilde{A_{m}}$. ∎

######  Remark 14.

Thus we have 8 proper normal subgroups in $S_{n}\wr{S_{m}}$ and 5 in
$A_{n}\wr{S_{m}}$ (active in the left). The list of new proper subgroups in
$E\wr S_{m}$, $S_{n}\wr\widetilde{A_{m}}$, $A_{n}\wr{S_{m}}$. Thus, we have 3
new proper normal subgroups.

In the next theorem active subgroup of wreath product is left.

######  Theorem 15.

There are exactly 7 normal subgroups in the wreath product $W=A_{n}\wr S_{n}$
(active is left), where 5 are proper normal. The list of these subgroups is
such:

1) $E$,

2) $E\wr\widetilde{A_{n}}$,

3) Subgroup of type $\widetilde{T_{n}}$ that is subgroup of the first level
stabilizer $St_{W}(1)$ whose elements having form
$g=(g_{1},g_{2},g_{3},\ldots,g_{n})$ then either all $g_{i}\in A_{n}$, or all
$g_{i}\in S_{n}\setminus A_{n}$,

For instance $W=S_{5}\wr A_{5}$ we have a subgroup of type $\widetilde{T_{n}}$
that is subgroup of $St(1)$ whose elements having form
$g=(g_{1},g_{2},g_{3},g_{4},g_{5})$ then either all $g_{i}\in A_{5}$, or all
$g_{i}\in S_{5}\setminus A_{5}$,

4) $St_{W^{\prime}}(1)$ this is subgroup of first type and has the structure
$E\wr\widetilde{A_{n}}$ (in other words it is subgroup of the first level
stabilizer $St_{W}(1)$),

5) $W^{\prime}$ this is subgroup of second type, where $W^{\prime}$ has the
structure $A_{n}\wr\widetilde{A_{n}}$. In general case if $W=A_{n}\wr A_{m}$
then $A_{n}\wr\widetilde{A_{m}}$ (active is on the left) note that $A_{n}\wr
A_{n}\lhd A_{n}\wr\widetilde{A_{n}}$, but $A_{n}\wr A_{n}$ is not normal in
$A_{n}\wr S_{n}$.

6) The level stabilizer $St_{W}(1)$ — proper normal subgroup of $W$. In this
case as well as in case if $W=S_{n}\wr S_{n}$ its normal subgroup $St_{W}(1)$
has structure $E\wr S_{n}$,

7) $W$ this is subgroup of second type.

All of these groups are split groups, 5 subgroups are proper subgroup.

###### Proof.

The proof of this theorem is written in the proof of Theorem 13 in terms of
Kaloujnine tableaux and normal closure of their generators. ∎

######  Remark 16.

A normalizer of ${{A}_{n}}\wr{{A}_{m}}$ in ${{S}_{n}}\wr{{S}_{m}}$ is the
following:

${{N}_{{{S}_{n}}\wr{{S}_{m}}}}({{A}_{n}}\wr{{A}_{m}})={{S}_{n}}\wr\widetilde{{{A}_{m}}}.$

Also we have to remark that
${{A}_{n}}\wr{{A}_{m}}\,\triangleleft\,{{S}_{n}}\wr{{A}_{m}}$.

Hypothesis. If $W={{S}_{n}}\wr{{\mathcal{A}}_{n}}$, where active subgroup is
left, then $W$ contains 4 proper normal subgroups and 6 normal subgroup. The
full list of the proper normal subgroups is the following:

1) $E\wr{{A}_{n}}$,

2) ${{A}_{n}}\wr{{A}_{n}}$

The proof is obvious.

Definition 9. A subgroup in $S_{n}\wr S_{n}\wr S_{n}$ is of the type
$\widetilde{T^{(3)}_{n^{2}}}$ if it consists of:

  1. 1.

elements of the form $E\wr E\wr A_{n}\,$,

  2. 2.

elements with the tableau [8] presentation $[e]_{1},\ [e\ldots,e]_{2},\
[\pi_{1}\ldots,\pi_{n};\pi_{n+1}\ldots,\pi_{n^{2}}]_{3}$, that $\pi_{i}\in
S_{n}\setminus A_{n}$.

Definition 10. A subgroup in $S_{n}\wr S_{n}\wr S_{n}$ is of the type
$\widetilde{T^{(3)}_{n}}$ if it has the structure:

$E\wr\widetilde{T_{n}}.$

######  Remark 17.

Note that vertex permutation of tableau third part satisfy the condition:

elements with the tableau [8] presentation $[e]_{1},\ [e\ldots,e]_{2},\
[\pi_{1}\ldots,\pi_{n};\pi_{n+1}\ldots,\pi_{n^{2}}]_{3}$, that either
$\pi_{i}\in S_{n}\setminus A_{n}$ or $\pi_{i}\in A_{n}$ and have the same
parity for $1<i<n$ also all elements with index $n+1<i<2n$ have identical
parity. The same is true for elements with indexes $kn+1<i<(k+1)n$, $k<n$.

Definition 11. (Subgroup $A^{(3)}_{n^{2}}$). The subgroup of $E\wr
E\wr{{A}_{n}}$ isomorphic to subdirect product of
${{A}_{n}}\times{{A}_{n}}\times{{A}_{n}}$, where permutations of third level
(${{X}^{3}}$) [24] in presentation of wreath product by automorphisms of
rooted tree acts in general on ${{X}^{4}}$ by even permutation is denoted by
$\tilde{A}_{{{n}^{2}}}^{(3)}$ or $E\wr E\wr{{\tilde{A}}_{n}}$. In another
words

$\prod\limits_{i=1}^{{{n}^{2}}}{{{\pi}_{3i}}\in{{A}_{{{n}^{2}}}}}$

The set of normal subgroup of ${{S}_{n}}\wr{{S}_{n}}$ is denoted by
$N({{S}_{n}}\wr{{S}_{n}})$. Subgroup with number $i$ from
$N({{S}_{n}}\wr{{S}_{n}})$ is denoted by ${{N}_{i}}({{S}_{n}}\wr{{S}_{n}})$.

######  Theorem 18.

The full list of normal subgroups of ${{S}_{n}}\wr{{S}_{n}}\wr{{S}_{n}}$
consists of 50 normal subgroups. These subgroups are the following:

  * 1

Type $T_{023}$ contains: $E\wr{{\tilde{A}}_{n}}\wr
H,\,\,\,\,\widetilde{T_{n}}\wr H$, where
$H\in\\{{{\tilde{A}}_{n}},{\tilde{A}}_{n^{2}},{{S}_{n}}\\}$. There are 6
subgroups.

  * 2

Additional class of subgroups in $T_{023}$ with new base of wreath product
subgroup ${{\tilde{A}}_{n^{2}}}$: $E\wr{{{S}}_{n}}\wr{{\tilde{A}}_{n^{2}}}$,
$E\wr{{N}_{i}}({{S}_{n}}\wr{{S}_{n}})$. Therefore this class has 12 new
subgroups. Thus, the total number of normal subgroups in Type $T_{023}$ is 18.

  * 3

Type $T_{003}$: $A_{00(n^{2})}^{(3)}=E\wr E\wr{{\tilde{A}}_{n^{2}}}$,
$\widetilde{T_{n^{2}}}^{(3)}$. Hence, here are 2 new subgroups.

  * 4

Type $T_{123}$: $\,\,{{N}_{i}}({{S}_{n}}\wr{{S}_{n}})\wr{{{S}}_{n}}$,
${{N}_{i}}({{S}_{n}}\wr{{{S}}_{n}})\wr{{\tilde{{A}}}_{n}}$ and
${{N}_{i}}({{S}_{n}}\wr{{{S}}_{n}})\wr{{\tilde{{A}}}_{n^{2}}}$. Thus, there 30
new subgroups in $T_{123}$.

Futhermore all of these groups are splitable.

Thus, we have find the complete list of 50 normal subgroups in
${{S}_{n}}\wr{{S}_{n}}\wr{{S}_{n}}$ there are 2 of them in non-proper
subgroups.

The proof is based on Theorem 13 stating that there are 10 normal subgroups in
$S_{n}\wr S_{n}$.

###  3.2 Normal subgroups in $\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$

Consider the set $N_{i}$ are the elements of a group in which the product of
permutations of each element (from any level) is an even substitution. This
set forms the subgroup $N_{i}$ of
$\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$. Here the index $i$ means the
smallest level at which a non-identical substitution occurs and calls the
depth of the subgroup $N_{i}$. Thus, the $i$-th level has a subdivision of the
tree T on which acts $\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$ on
$\displaystyle\prod_{i=1}^{j}n_{i}$ subtrees, at each level of which the
product of substitutions is even.

Statement. The following subgroup embedding is valid:
$N_{1}\supset N_{2}\supset N_{3}\supset...$

Proof. Let $1\leq m\leq k$ then at the $i-$th level, where $m\leq i\leq k$, in
the elements of $N_{k}$ are identical substitutions $e$. Therefore, their
product at the $m$-th level is an even permutations, which is a characteristic
property of $N_{m}$, so $N_{k}$ belongs to $N_{m}$.

Note that is, $N_{m}$ is a group. The definition of
$\S_{i=1}^{\infty}S_{n_{i}}$ implies that the product of its elements from the
$k$-th level forms as the product of permutations of elements the $k$-th
level. It means that product of two element with the even product (3) is an
element satisfying the property (3) too.

The presence of the inverse element in such a set $N_{m}$ can be proved by
contradiction.

######  Definition 11.

Depth of element We call by $k$-coordinate tableau such tableau having
${{\left[u\right]}_{l}}=\varepsilon$ when $l\neq k$. The subgroup of all
$l$-coordinate tableaux is denoted by ${{\left[W\right]}_{l}}$.

######  Theorem 19.

The bases $B_{i}$ forms topology of wreath product by determining of open
neighborhoods system of neutral element —
$t=[\varepsilon,\varepsilon,\,...\,,\,\varepsilon,...]$.

######  Definition 12.

Tableau $u$ has depth $m$ if all ${{[u]}_{i}}=\varepsilon$ provided
$i=1,2,\,\,...\,,\,m-1$; and ${{[u]}_{m}}\neq\varepsilon$. We set that depth
of $e$ is equal to $\infty$.

Statement. Each subgroup $N_{i}$ is a normal subgroup in
$\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$. Furthermore $N_{i}$ form the
$i$-th base of wreath product $\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$.

Obviously, because the conjugation at each level can only add an even number
of odd permutations. (Moreover, the conjugation does not change the cyclic
type of substitution). Each substitution of the $i$ 1st level conjugates only
one substitution of the same level, therefore the rule (4) is applicable.

In general case conjugation by arbitrary element does not change parity of
product of elements of $N_{i}$.

######  Theorem 20.

Let $H$ be a normal subgroup of
$W=\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$. Let us assume that $N_{1}<H$.
Then $\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}/H$ is isomorphic to some
subgroup of the group $\displaystyle\prod_{1}^{\infty}$ $\mathbb{Z}_{2}$.

###### Proof.

Recall that we have defined subgroup of $l$-th level $W(l)$ in [24] of the
group $W$.

Define a homomorphism from $W(l)$ onto $Z_{2}$, whose kernel is $H(l)$.
Another words kernel of this homomorphism is subgroup $H(l)$ of l-th level of
group $H(l)$. If products of transpositions that belongs to $H(l)$ is even
then image of $W(l)$ is non trivial element 1 from ${{Z}_{2}}$, if products of
its transpositions is odd permutation then its image is 0.

The proof will be constructed by adding elements to the group $N<H$. Since,
$N$ contains arbitrary element that having an even product of level
permutations, we need to add those elements that are characterized only by
product of permutations on each level. We are interested only in existence of
odd permutation on each level, because by conjugation it is possible to place
permutations of arbitrarily level in arbitrarily order. Let us consider two
arbitrarily elements $a,b\in\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$. Let
us also consider $l-th$ level of each of these elements. Product of
permutations in $a^{-1}ba$ of $l-th$ level is even if and only if their
product as elements of a group $S_{n_{l}}$ forms even permutation of
$S_{n_{l}}$. In general case, we have to notice that conjugation by an
arbitrary element cannot change the product of permutations on each level each
sub-tree, that corresponds to $H$. Indeed, by the definition in
$\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$ each permutation $i+1$-th level
conjugate one permutation of $i+1$-th level of initial element. Therefore, all
$H$ are normal subgroups in $\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$.

Let’s consider bijection
$\phi:\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}\rightarrow\displaystyle\prod_{1}^{\infty}$
$\mathbb{Z}_{2}$. In other words, $\phi$ gives to each element from
$\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$ infinite sequence of zeros and
ones. This bijection is homomorphism of the group. If we assume that all
elements of the group $\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$ are
equivalent if and only if parity of products of elements in all levels
coincides. Since, each representative of the class will correspond to only one
element from $\displaystyle\prod_{1}^{\infty}$ $\mathbb{Z}_{2}$, we get
statement of the theorem. ∎

######  Theorem 21.

Normal subgroups, that consist of finite elements, form a lattice that is
isomorphic to a lattice of subgroups of infinite iterate in one side Cartesian
product of cyclic groups of order 2.

###### Proof.

Proof is obvious from the previous $Theorem$. ∎

######  Theorem 22.

Each $N_{i}$ is a normal subgroup of
$\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$.

###### Proof.

We need to prove that conjugation of elements from $N_{i}$ with elements from
$\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$ belongs to $N_{i}$. Consider
$N_{1}$. In this case everything is obvious: on each level conjugation can add
only even number of odd permutations, that will not change the parity of the
product of permutations on each level. In general case, we have to notice that
conjugation by an arbitrary element cannot change the product of permutations
on each level each sub-tree, that corresponds to $N_{i}$. Indeed, by the
definition in $\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$ each permutation
$i+1$-th level conjugate one permutation of $i+1$-th level of initial element.
Therefore, all $N_{i}$ are normal subgroups in
$\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$.

Let $(\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}))_{l-1}$ is a subgroup of
$\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$, where identical permutations
are on the first $l-1$ levels. The following theorem holds:
Consider some matrix space $M$ over $\mathbb{Z}_{2}$, each matrix has infinite
number of rows. We assign to each depth $l$ of some element $\mathbb{Z}_{2}$
the number $\prod_{i=1}^{l-1}n_{i}$.

Definition Two elements of matrix space $M$, that is characterized by a
integer $l$ and group $\mathop{\wr}\limits_{i=1}^{\infty}S_{n_{i}}$, is
$\omega$-equivalent, if one of these elements can be obtained from another by
the following transformations:
1\. aloud to permute first $n_{l-1}$ columns, second $n_{l-1}$ columns, etc.
2\. aloud to permute first $n_{l}-2$ groups of columns, that consist of
$n_{l-1}$ columns, second $n_{l-2}$ groups of columns , etc.
3\. aloud to permute first $n$ groups of columns. ∎

##  4 Inverse limit of wreath product of permutation groups

_Definition._ Baer space is defined as the Cartesian product of an infinitely
countable set of copies of a set of natural numbers and endowed with a product
topology (where each copy of the set of natural numbers is assigned a
_discrete topology_).

Baire space is often presented by a tree of finite sequences of natural
numbers.

Let ${{W}_{1}}={{S}_{X}}$ symmetric group of permutations on $X$, then we can
determine group recursively ${{W}_{n+1}}=({{W}_{n}},{{X}^{n}})\wr{{S}_{X}}$ as
wreath product of ${{W}_{n}}$ acting of ${{X}^{n}}$ with permutation group
acting on $X$. The base of wreath product consists of functions
$Fun({{X}^{n}},\,{{S}_{X}})$. But taking into account that ${{S}_{X}}$ acts on
$X$ we can consider functions $Fun({{X}^{n}},\,X)=\,:B_{n}$. This subgroup $B$
is kernel of natural epimorphism: ${{\rho}_{n}}:{{W}_{n+1}}\to{{W}_{n}}$. Or
in general case ${{\rho}_{k,n}}:{{W}_{k}}\to{{W}_{n}}$. In such a way we can
construct inverse system of groups:

$I=\left\langle{{\rho}_{n}},\,\,{{W}_{n+1}}\right\rangle$

In general case we can consider different alphabets ${{X}_{i}}$ then
$\left\langle{{\rho}_{n}},\,\,{{W}_{n+1}}\right\rangle=\left\langle{{\rho}_{n}},\,\,\underset{i=1}{\overset{n+1}{\mathop{\wr}}}\,{{S}_{{{n}_{i}}}}\right\rangle$
This projective limit we will denote by

$P{{W}_{\infty}}\left(X\right):=\underleftarrow{\lim}\,{{W}_{n}}.$

We define canonical projection as the following homomorphism from projective
limit to objects of inverse system:
${{\pi}_{n}}:P{{W}_{\infty}}(X)\to{{W}_{n}}(X)$. Provided this projection, the
group $P{{W}_{\infty}}\left(X\right)$ acts on Baire space over alphabet $X$.

The group $P{{W}_{\infty}}(X)$ is the projective limit of $I$ because of:

1) there exist set of mappings
${{\pi}_{n}}:P{{W}_{\infty}}(X)\to{{W}_{n}}({{X}^{n}})$,
${{\pi}_{k}}:P{{W}_{\infty}}(X)\to{{W}_{k}}({{X}^{k}})$ and epimorphism
${{\rho}_{n,k}}:{{W}_{k}}({{X}^{k}})\to{{W}_{n}}({{X}^{n}}),\,\,\,\forall
n\leq k$ such that ${{\pi}_{n}}={{\rho}_{n,k}}{{\pi}_{k}}$, where
${{\pi}_{i}}$ is canonical projection on i-th component of algebraic system of
groups ${{W}_{i}}(X)$. The mapping
${{\rho}_{n}}:{{W}_{n+1}(X)}\to{{W}_{n}(X)}$ is epimorphism with the kernel
$\ker({{\rho}_{n}})=Fun({{X}^{n}},{{S}_{X}})$.

2) Also the following composition of epimorphism is epimorphism again:
${{\rho}_{n,m}}={{\rho}_{n,k}}{{\rho}_{k,m}},\,\,\,\forall n\leq k\leq m$.

We consider the bases ${{B}_{i}}$ as open neighborhoods system of neutral
element $e$ of $W=\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$ that is a
normal subgroup in $\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$. Furthermore
$B_{i}$ form the $i$-th base of wreath product
$\mathop{\wr}\limits^{\infty}_{i=1}S_{n_{i}}$.

For this goal firstly open neighborhoods ${{B}_{i}}$ have to satisfy to
conditions of there must be consistency over multiplication (or an action on
an open set) $g{{B}_{i}}={{B}_{j}}$ for $\forall i\in\,\mathbb{N}$.

Secondly for an arbitrary open neighborhoods ${{B}_{i}}$ of $e$ there must
exists an inverse element $B_{i}^{-1}$ in the group $W$.

It is obviously that these conditions are satisfied for bases $B_{i}$ of $W$.

We call by $k$-coordinate tableau such tableau having
${{\left[u\right]}_{k}}=\varepsilon$ when $k\neq l$. The subgroup of all
$l$-coordinate tableaux is denoted by ${{\left[W\right]}_{l}}$.

Tableau $u$ has depth $m$ if all ${{[u]}_{i}}=\varepsilon$ provided
$i=1,2,\,\,...\,,\,m$; and ${{[u]}_{i}}\neq\varepsilon$.

######  Remark 23.

The subgroup ${{\left[W\right]}_{l}}$ is isomorphic to ${{\mu}_{l}}$-th direct
power of subgroup ${{S}_{l}}$ from wreath product
$W_{l}={{S}_{{{X}_{1}}}}\wr{{S}_{{{X}_{2}}}}\wr\,\,.\,.\,.\,\,\wr{{S}_{{{X}_{l}}}}\wr{{S}_{{{X}_{l+1}}}}$,
where
${{\mu}_{l-1}}=\left|{{X}_{1}}\times{{X}_{2}}\times...\times{{X}_{l-1}}\right|$.

Conclusion. Further, the monolith (i.e. the unique minimal normal subgroup) of
these wreath products has been investigated. The monolith exists when $n\equiv
0(mod2)$.

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11institutetext:  Division of Medical Image Computing, German Cancer Research
Center, Heidelberg, Germany 22institutetext: HIP Applied Computer Vision Lab,
Division of Medical Image Computing, German Cancer Research Center
33institutetext: Interactive Machine Learning Group, German Cancer Research
Center 44institutetext: Department of Neuroradiology, Heidelberg University
Hospital, Heidelberg, Germany 55institutetext: Neurology Clinic, Heidelberg
University Hospital 66institutetext: DKTK CCU Neurooncology, German Cancer
Research Center 77institutetext: Division of Molecular and Translational
Radiation Oncology, Heidelberg Institute of Radiation Oncology (HIRO),
Heidelberg, Germany 88institutetext: Heidelberg Ion-Beam Therapy Center (HIT),
Heidelberg University Hospital 99institutetext: Clinical Cooperation Unit
Radiation Oncology, German Cancer Research Center 99email:
<EMAIL_ADDRESS>
# Continuous-Time Deep Glioma Growth Models

Jens Petersen 11 Fabian Isensee 22 Gregor Köhler Paul F. Jäger 1133 David
Zimmerer 11 Ulf Neuberger 44 Wolfgang Wick 5566 Jürgen Debus 778899 Sabine
Heiland 44 Martin Bendszus 44 Philipp Vollmuth 44 Klaus H. Maier-Hein 11

###### Abstract

The ability to estimate how a tumor might evolve in the future could have
tremendous clinical benefits, from improved treatment decisions to better dose
distribution in radiation therapy. Recent work has approached the glioma
growth modeling problem via deep learning and variational inference, thus
learning growth dynamics entirely from a real patient data distribution. So
far, this approach was constrained to predefined image acquisition intervals
and sequences of fixed length, which limits its applicability in more
realistic scenarios. We overcome these limitations by extending Neural
Processes, a class of conditional generative models for stochastic time
series, with a hierarchical multi-scale representation encoding including a
spatio-temporal attention mechanism. The result is a learned growth model that
can be conditioned on an arbitrary number of observations, and that can
produce a distribution of temporally consistent growth trajectories on a
continuous time axis. On a dataset of 379 patients, the approach successfully
captures both global and finer-grained variations in the images, exhibiting
superior performance compared to other learned growth models.

###### Keywords:

Glioma Growth Stochastic Time Series Generative Modeling

##  1 Introduction

Glioma growth modeling refers to the problem of estimating how the tumor
burden in a patient might evolve in the future given the current state of the
disease and possibly its history. In the past, this was often approached with
so-called reaction-diffusion models, where tumor growth is described by the
physical diffusion of an assumed tumor cell density. There is a large body of
related literature, so we only refer to [16, 17] as overviews. Newer works not
included therein are [20, 4], among others. All of these approaches have in
common that they depend on the quality of the chosen forward model, which
always includes simplifying assumptions when compared to the actual biological
processes.

A recent alternative was proposed in [18] in the form of fully learned growth
models. The authors used a probabilistic U-Net to map a fixed-length input
sequence of MRI scans to a distribution of possible spatial tumor
configurations in the future. The model thus learns a growth model directly
from data and doesn’t rely on explicit assumptions and simplifications with
respect to the underlying biological process. At the same time, the approach
suffers from several limitations, requiring a fixed number of input
observations, requiring fixed time intervals between observations, and being
able to make predictions only for a single predefined future time point. In
order to overcome all of these limitations, we build upon the Neural Process
framework [5, 6], a family of conditional generative models specifically
conceived to model continuous-valued stochastic processes. They work by
encoding so-called _context_ observations into a joint representation and then
predicting a desired _target_ output from it. In particular, we extend the
work from [3, 13], who apply Neural Processes in the image domain, by
introducing a convolutional encoder-decoder structure with representations at
multiple scales. The single-scale encoder-decoder setups used in [3, 13]
require extremely large decoders to produce outputs with sufficient spatial
resolution. As a result, they require vast amounts of data to be trained
successfully, and the authors only demonstrate their approaches on synthetic
images. Ours is one of the first Neural Processes to be applied to real data
in the image domain111 _Image domain_ refers to the fact that the observations
are entire images. The original Neural Processes [5, 6] work on images by
treating individual pixels as observations., with [8] the only other work
we’re aware of. To allow our approach to model tumor changes with high spatial
resolution, we also introduce an attention mechanism [21] that aggregates the
context information at different scales. In contrast to existing work with
attention in Neural Processes [10, 19], we design the mechanism so that it can
model spatial and temporal dependencies simultaneously.

In summary, this work makes the following contributions: 1.) We propose the
first fully learned tumor growth model that can make predictions at arbitrary
points on the time axis, conditioned on an arbitrary number of available
observations. 2.) In doing so, we propose a Neural Process variant for image-
based problems that is efficient enough to work successfully on real-world
data. 3.) We show that the proposed approach significantly outperforms prior
work on the task of learning-based modeling of glioma growth. We compare our
approach with both the model proposed in [18] and a naive convolutional Neural
Process. 4.) We provide a complete implementation at https://github.com/MIC-
DKFZ/deep-glioma-growth.

Figure 1: A schematic of the proposed model. All _context_ observations
(MRI/segmentation/time, blue) are encoded separately with the same
convolutional encoder that outputs representations at different resolution
scales. The lowest scale (black) has no spatial resolution, and the
corresponding representations are summed. The result is a global distribution,
a sample of which is fed to the decoder together with the _target_ input
(time, red). For representations with low spatial resolution, we use spatio-
temporal attention (SA) to aggregate context information. At higher
resolutions we use only temporal attention (TA) to reduce GPU memory
requirements. ConcatCoords represents an operator that appends pixel locations
($-0.5$ to $0.5$ per axis). The first encoder block also uses InstanceNorm.

##  2 Methods

###  2.1 Model

The model we propose is shown in Fig. 1. We have a number of observations
(MRIs and corresponding tumor segmentations) available at continuous time
values $\bm{t_{c}}$. We call those the _context_ , with bold indicating a
collection henceforth. We are interested in possible tumor segmentations for
future _target_ times $\bm{t_{t}}$. Each context observation is encoded
separately with an encoder that outputs representations hierarchically at
multiple scales:
$\bm{r_{c}=(r_{c,32},r_{c,16},r_{c,8},r_{c,4},r_{c,global})}$. The second
index refers to the scale (the first entry has a resolution of $32^{2}$, etc.)
and the global representations $\bm{r_{c,global}}$ describe mean and standard
deviation of diagonal normal distributions. These means and standard
deviations are summed over all context encodings (bottom path in Fig. 1) to
form a single global distribution. Because it is independent of the target
$t_{t}$, this allows us to sample growth trajectories that are consistent over
time and that can be evaluated at arbitrary new points in time. A sample from
the distribution (which is 128-dimensional in our case) along with $t_{t}$ is
fed to the decoder to predict the corresponding segmentation. If only the
global representation is used, our model reduces to a Neural Process [5, 6]
with a convolutional encoder and decoder. We want our approach to be able to
model spatial and temporal dependencies simultaneously, so we aggregate the
context information at higher scales using a spatio-temporal dot-product
attention mechanism [21], conditioned on the target $t_{target}$ and pixel
locations $(x,y)$ at the given scale (see inset in Fig. 1):

$\displaystyle
r(\bm{t_{t}},\bm{x_{t}},\bm{y_{t}})=\mathrm{softmax}\left(\frac{\bm{QK}^{T}}{\sqrt{d}}\right)\bm{V}$
(1)
$\displaystyle\bm{Q}=\mathrm{Lin}\,(\bm{t_{t}},\bm{x_{t}},\bm{y_{t}},\bm{r_{c,x,y}})\;,\;\bm{K}=\mathrm{Lin}\,(\bm{t_{c}},\bm{x_{c}},\bm{y_{c}},\bm{r_{c,x,y}})\;,\;\bm{V}=\mathrm{Lin}\,(\bm{r_{c,x,y}})$
(2)

$\bm{Q},\bm{K},\bm{V}$ are linear maps from what is usually referred to as
_queries_ (target locations $\bm{(t_{t},x_{t},y_{t})}$), _keys_ (available
locations $\bm{(t_{c},x_{c},y_{c})}$) and _values_ (the representations at the
_key_ locations). We also use $\bm{r_{c,x,y}}$ to construct $\bm{Q}$ and
$\bm{K}$ because of the success of self-attention in transformer
architectures, e.g. [1]. This means that the available representations are
weighted according to the similarity of the corresponding key to the queries.
The similarity is given by the dot product in a higher-dimensional space (with
dimension $d=16$ in our case). Note that in Fig. 1, each attention block
represents so-called multihead attention, meaning 8 parallel blocks of the
above. The downside of this spatio-temporal attention is that the matrix
product $\bm{QK^{T}}$ requires GPU memory that scales quadratically with the
number of pixels. As a result we only use the full spatio-temporal attention
at the two lowest scales (resolutions $4^{2}$ and $8^{2}$) and resort to
temporal attention at higher scales, i.e. the same as above but without a
dependence on pixel locations.

Our model is trained with the following optimization objective, where $I$ are
inputs (MRI scans) while $S$ refers to segmentations. During training the
context is absorbed into the target set ($\bm{t_{t}}\rightarrow\bm{t_{t+c}}$,
etc.), meaning we let the models reconstruct the inputs as well instead of
only predicting a future segmentation:

$\displaystyle\min_{\theta}\mathop{\mathbb{E}}_{z\sim
q_{\theta}(z|\bm{I_{t}},\bm{S_{t}},\bm{t_{t}})}$ $\displaystyle-\log
p_{\theta}(\bm{S_{t}}|\bm{I_{c}},\bm{S_{c}},\bm{t_{c}},\bm{t_{t}},z)$
$\displaystyle+\beta\cdot
D_{KL}(q_{\theta}(z|\bm{I_{t}},\bm{S_{t}},\bm{t_{t}})||q_{\theta}(z|\bm{I_{c}},\bm{S_{c}},\bm{t_{c}}))$
(3)

where $q_{\theta}(z|\cdot)$ is the global representation predicted by the
encoder for the given inputs. This is the Neural Process variant of the well-
known variational inference objective, with some modifications: 1.) There is a
direct dependence of the reconstruction likelihood $p_{\theta}$ on
$(\bm{I_{c},S_{c},t_{c}})$, because the attention mechanisms are deterministic
(otherwise the influence would only be via $z$). 2.) The predictive likelihood
is usually modeled as a factorizing categorical likelihood, leading to a
cross-entropy loss. We instead use the sum of cross-entropy and Dice losses
[2]. As a result, the likelihood no longer factorizes over pixels, so 3.) we
introduce a weight $\beta$ for the KL term, as done for example in [7], which
re-balances the two loss terms. We optimized this parameter through line-
searching the space of possible values in powers of 10, choosing the one that
lead to KL loss values most similar to those obtained during training with
only cross-entropy: $\beta=0.0001$. Note that the objective for the baseline
from [18] can be written the same way, but without dependence on
$(\bm{S_{c},t_{c},t_{t}})$. Models are trained for 300 epochs with Adam [11]
and a learning rate of $0.0001$ at batch size 128. We give between 2 and 5
inputs as context and always have a single target point, but the models
predict both context and target during training. For all other hyperparameters
we refer to the provided implementation.

###  2.2 Data & Evaluation

We use a dataset that consists of MRI scans from 379 glioblastoma patients,
stemming from a multi-center study that compared treatment using a combination
of chemotherapy (Lomustine) and an angiogenesis inhibitor (Bevacizumab) with
treatment using chemotherapy alone [23]. The study found no significant
difference between test and control arm in terms of overall survival, so we
ignore treatment effects in our work and treat it as an additional source of
stochasticity. Pre-processing of the data including annotations (segmentations
of edema, enhancing tumor and necrosis) was performed by experienced
neuroradiologists as described previously [9]. For each patient there are
between 3 and 13 (mean $4.85$) longitudinal scans, consisting of native T1,
contrast-enhanced T1 (with gadolinium agent), T2 and FLAIR. All scans for a
patient are skull-stripped, registered to the native T1 image of the first
available scan, and normalized by mean and standard deviation on a per-channel
basis. We resample the original data to isotropic $128^{3}$ and then extract a
region of size $64^{3}$ centered around the tumor. We find that this offers a
good trade-off between resolution and compute requirements. Finally, we work
on axial slices of the resulting arrays, making extensive use of data
augmentation during training. A full list of transforms is given in the
implementation. Note that our model only uses skip connections with a
resolution of at most $32^{2}$. For the evaluation we perform a random 80/20
train/test split of the data (stratified at the patient level to avoid data
leakage). Hyperparameter optimization is done via cross-validation on the
training set, for the final results we train all models on the full training
set and report scores on the test set, leaving out those instances where the
true Dice overlap between the target point and the last context point is 0.

We compare our approach with the one presented in [18] using their official
implementation and a hyperparameter configuration that matches our model
(U-Net layer size and depth, loss, etc.). We further compare with a regular
Neural Process [6], which is a special case of our model where all the
attention skip-connections are removed (see Fig. 1). Because the approach in
[18] works with a fixed number of input timesteps, we train 4 separate models
(2-5 input timesteps) and evaluate them jointly. We look at three different
metrics: 1) The _Test Loss_ , which is just the optimization objective (Eq. 3)
evaluated on the test set, but only using the test points and not the context
reconstructions. It is by definition the optimal measure of how well the
models perform the task they are trained to perform. Note that this measure is
only meaningful when all models are trained with the same loss, which we do.
2) The _Surprise_ is the KL divergence, i.e. the second summand in Eq. 3. It
can be interpreted as a measure of how much the model has to adjust its prior
belief when presented with the true future observation (the posterior). 3) The
_Query Volume Dice_ takes 100 random samples from the predicted prior and
measures the Dice overlap for the sample that best matches the observed future
in terms of whole tumor volume. It is best interpreted from an application
perspective, when clinicians are interested in possible spatial growth
estimates _conditioned_ on a potential volume increase. Note that we are
mainly interested in a comparison with other learned growth models, and the
chosen metrics are not suitable for a comparison with diffusion-based models.
We leave such an evaluation for future work.

Figure 2: Example case with predictions from all methods we compare in this work (purple=edema, orange=enhancing tumor, yellow=necrosis). Red borders mark the context, we only show the T1c input channel. Our model can reconstruct the context observations very accurately, while the Neural Process prediction lacks spatial resolution. The mean prediction of our model anticipates an expansion of the necrosis in the tumor center (at $t=265$ days), which is commonly observed, but does not occur in this particular instance. The query volume prediction is very close to the observed future, indicating that our model does indeed learn meaningful growth trajectories. The model from [18] can only make a prediction at a single fixed point in time. Table 1: Results on the test set. $\uparrow/\downarrow$ indicate that higher/lower is better, bold marks the best-performing method. Errors represent the standard error of the mean. _Test Loss_ evaluates Eq. 3 on the test set, _Surprise_ measures the distance between prior and posterior distributions. _Query Volume Dice_ draws 100 samples from the prior and evaluates the Dice overlap for the sample that is closest to the true future segmentation in terms of tumor volume. Our model outperforms the baselines in all metrics. |  | Test Loss $\downarrow$ | Surprise $\downarrow$ | Query Vol. Dice $\uparrow$
---|---|---|---|---
| Parameters | [1e-2] | [nats] | [1e-2]
Neural Process [6] |  $5.9$M | $34.1\pm 0.2$ | $431.9\pm 3.7$ | $57.5\pm 0.2$
Learned Discrete [18] |  $11.2$M | $38.9\pm 0.3$ | $112.1\pm 0.7$ | $60.1\pm 0.2$
Learned Continuous (ours) |  $6.4$M | $24.1\pm 0.2$ | $82.7\pm 1.5$ | $71.4\pm 0.2$
Figure 3: _Predictive Dice_ and _Query Volume Dice_ at different thresholds of
true overlap. There is often very little change between consecutive
observations, so we filter test cases by varying a threshold of their true
overlap (Dice) to see how performance changes. As an example, if we select a
threshold of 0.6, the corresponding y-values are the average performances on
test cases where the overlap of the test timestep with the last context input
is less than 0.6. The _predictive threshold_ is the average true overlap of
those thresholded cases (and the average true Dice for all test cases is
$0.67$), meaning that if a model falls below that line, it would be better to
just assume no change will take place instead of trusting the model’s
prediction. Only our proposed model stays above that line for all thresholds.

##  3 Results

An exemplary case is shown in Fig. 2, with predictions from all models. We
find that our model produces predictions with high spatial resolution. It can
reconstruct the context segmentations very accurately, while the Neural
Process is unable to create sufficient spatial detail. The mean prediction
from our model overestimates how fast the necrosis (yellow in Fig. 2) in the
tumor center expands, but it is important to consider that the observed future
isn’t necessarily the most likely one, so it is expected to find such
differences. More importantly, when we sample from the predicted prior and
take the samples that are closest to the observed future in terms of tumor
volume, our model matches the observed future closely. The model from [18] can
produce spatial detail comparable to our model, but it can only make a
prediction at a single fixed time relative to the context observations.

The average performance of all methods on the test set is shown in Table 1. We
evaluate the _Test Loss_ (i.e. Eq. 3, but only on target points, not context
reconstructions), the _Surprise_ , meaning the KL divergence between prior and
posterior, as well as the _Query Volume Dice_ , which is the Dice for the
sample that best matches the observed future in terms of tumor volume (out of
100 samples from the prior). Our method significantly outperforms the
competing approaches in all metrics, with only a little more than half the
number of parameters that the model from [18] uses, because ours doesn’t
require separate prior and posterior networks. Interestingly, the test loss of
the Neural Process [6] is actually lower than that of the learned discrete
model [18], even though its perceived quality is much lower (see Fig. 2). It
should be noted that the numerical results are not comparable to those
reported in [18], because we use a slightly different training objective.
There are many test cases with only very small changes between the test
timestep and the previous context observation. To understand how the
performance depends on changes that take place between those times, we iterate
a threshold over their true Dice overlap to evaluate the average _Predictive
Dice_ and Query Volume Dice on the subset of cases below the given threshold.
The result is displayed in Fig. 3, where we also include a predictive
threshold, which is the average true overlap below the given threshold. Below
that line, a model performs worse than one that simply predicts no change
between consecutive times. Ours is the only approach that is always above the
predictive threshold, while the others actually fall _below_ it at higher
cutoffs.

##  4 Discussion

This work proposes a fully learned glioma growth model, first introduced in
[18] as an alternative to commonly used biological diffusion models [16, 17].
While successful, the approach from [18] has a number of practical
limitations: it requires a fixed number of context observations, it requires a
fixed time interval between consecutive observations and it can only make a
prediction one time interval step into the future. Our proposed model
overcomes all of those limitations and can be conditioned on arbitrarily many
context observations on a continuous time axis. From these context
observations, our model predicts a distribution over growth trajectories,
where each sample is temporally consistent and can be evaluated at any desired
continuous-valued time. Our model also significantly outperforms the one from
[18] in several metrics, which we demonstrate on a dataset ten times larger
than the one used in [18]. Our model’s main limitation is the high GPU memory
requirement of the spatio-temporal attention mechanism we introduce. This is a
problem many attention and transformer architectures suffer from, and ways to
make them less resource intensive are actively being researched [12, 22]. It’s
also the reason why we performed our experiments on two-dimensional slices
rather than full 3D volumes. As a result, one should be careful not to draw
any premature conclusions with respect to a possible clinical application.
While our results look promising, the value of our model for clinical
purposes, for example in radiation therapy, must be validated extensively. We
leave such efforts for future work. A comparison with diffusion-based models,
particularly in a radiation therapy context [15, 14], is another interesting
opportunity for future work. Building on the Neural Process framework [5, 6],
our proposed approach constitutes an efficient Neural Process variant for
image time series and is, to the best of our knowledge, only the second time
Neural Processes have been demonstrated on real data in the image domain [8].
We believe it can prove useful for both other types of tumor growth as well as
any other kind of stochastic time series with image data.

## Acknowledgements

Part of this work was funded by the Helmholtz Imaging Platform (HIP), a
platform of the Helmholtz Incubator on Information and Data Science.

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