Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees
The paper concerns the Sylow p-subgroups of automorphism groups of level homogeneous rooted trees. We recall and summarize the results obtained by L.Kaluzhnin on the structure of Sylow p-subgroups of isometry groups of ultrametric Cantor p-spaces in terms of automorphism groups of rooted trees. Mos...
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irk-123456789-1527832019-06-13T01:25:07Z Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees Bier, A. Sushchansky, V. The paper concerns the Sylow p-subgroups of automorphism groups of level homogeneous rooted trees. We recall and summarize the results obtained by L.Kaluzhnin on the structure of Sylow p-subgroups of isometry groups of ultrametric Cantor p-spaces in terms of automorphism groups of rooted trees. Most of the paper should be viewed as a systematic topical survey, however we include some new ideas in last sections. 2015 Article Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees / A. Bier, V. Sushchansky // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 19-38. — Бібліогр.: 22 назв. — англ. 1726-3255 2010 MSC:20B27, 20E08, 20B22, 20B35, 20F65, 20B07. http://dspace.nbuv.gov.ua/handle/123456789/152783 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The paper concerns the Sylow p-subgroups of automorphism groups of level homogeneous rooted trees. We recall and summarize the results obtained by L.Kaluzhnin on the structure of Sylow p-subgroups of isometry groups of ultrametric Cantor p-spaces in terms of automorphism groups of rooted trees. Most of the paper should be viewed as a systematic topical survey, however we include some new ideas in last sections. |
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Bier, A. Sushchansky, V. |
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Bier, A. Sushchansky, V. Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees Algebra and Discrete Mathematics |
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Bier, A. Sushchansky, V. |
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Bier, A. |
| title |
Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees |
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Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees |
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Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees |
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Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees |
| title_full_unstemmed |
Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees |
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kaluzhnin's representations of sylow p-subgroups of automorphism groups of p-adic rooted trees |
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Інститут прикладної математики і механіки НАН України |
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2015 |
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Kaluzhnin's representations of Sylow p-subgroups of automorphism groups of p-adic rooted trees / A. Bier, V. Sushchansky // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 19-38. — Бібліогр.: 22 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT biera kaluzhninsrepresentationsofsylowpsubgroupsofautomorphismgroupsofpadicrootedtrees AT sushchanskyv kaluzhninsrepresentationsofsylowpsubgroupsofautomorphismgroupsofpadicrootedtrees |
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2025-07-14T04:16:43Z |
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2025-07-14T04:16:43Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 19 (2015). Number 1, pp. 19–38
© Journal “Algebra and Discrete Mathematics”
Kaluzhnin’s representations
of Sylow p-subgroups of automorphism groups
of p-adic rooted trees
Agnieszka Bier and Vitaliy Sushchansky
Abstract. The paper concerns the Sylow p-subgroups of
automorphism groups of level homogeneous rooted trees. We recall
and summarize the results obtained by L. Kaluzhnin on the structure
of Sylow p-subgroups of isometry groups of ultrametric Cantor
p-spaces in terms of automorphism groups of rooted trees. Most of
the paper should be viewed as a systematic topical survey, however
we include some new ideas in last sections.
1. Introduction
The Sylow p-subgroup P∞ of the automorphism group of a p-adic
rooted tree is one of the most popular examples of pro-p groups.This is
due to the universality of P∞ in the classes of all countable residually-p
groups and profinite groups of countable weight. The group has natural
characterizations in topological terms (as the group of isometries of Cantor
metric space), geometrical terms (as a group of automorphisms of a rooted
tree) and algebraic terms (as an infinitely iterated wreath product of
cyclic groups of order p). It may be also defined as the limit of the inverse
spectrum of Sylow p-subgroups in symmetric groups of orders p, p2, p2,...,
etc. If p = 2 the group P∞ coincides with the whole automorphism group
of the binary tree.
2010 MSC: 20B27, 20E08, 20B22, 20B35, 20F65, 20B07.
Key words and phrases: Sylow p-subgroups of wreath products of groups,
homogeneous rooted trees, automorphisms of trees.
20 Kaluzhnin’s representations
It is worth mentioning that all known examples of residually finite
groups of Burnside type may be embedded in the Sylow p-subgroup P∞
of the automorphism group of a p-adic rooted tree, and most of them were
obtained just as subgroups of P∞. Therefore P∞ remains an interesting
object of investigations and important for the overall theory.
The group P∞ has been introduced almost 70 years ago by L. Kaluzh-
nin in his note [12] as a generalisation of the finite groups Pm, more
precisely as an inverse limit of the system of groups Pm, m ∈ N. The
note was a part of a series of papers on Sylow p-subgroups of symmet-
ric groups [8–14]. In his paper [14] professor Kaluzhnin summarized his
results on P∞. The main of these are:
(i) characterization of P∞ as a Sylow p-subgroup of isometry group of
an ultrametric Cantor p-space, metrization of P∞ and investigations
of its universal properties among so called p∞-groups;
(ii) introduction of the concept of parallelotopic subgroups and deter-
mination of necessary and sufficient conditions for these subgroups
to be normal in P∞;
(iii) characterization of invariant to isometric automorphisms subgroups
of P∞.
The results (i)–(iii) were obtained as generalisations of the respective
results in groups Pm. After Kaluzhnin’s papers the group structure of
P∞ has not been much investigated. Some results concerning the au-
tomorphism group of P∞ were obtained by P. Lentoudis in [20], who
studied isometric automorphisms and gave their complete description.
In [1] M. Abert and B. Virag discussed some probabilistic properties
of groups acting on rooted trees. In [22] V. Sushchansky discussed em-
beddings of residually finite p-groups into the groups of isometries of
the space of p-adic integers. Some facts regarding this group follow from
investigations of infinitely iterated wreath products of groups [15], branch
groups [2] and some others [18,19].
In the presented paper we give a systematic overview on the results
known on the structure of the group P∞ and provide a few new facts in
this topic. In Section 2 we introduce the notion of a (finite) rooted p-tree
and discuss the group of its automorphisms. Section 3 concerns automor-
phism groups of infinite p-trees and provides a useful representation of
elements of such groups. In Section 4 we discuss the standard embedding
of residually finite groups as topological groups (the coset representation)
into the automorphism groups of some infinite level homogeneous tree.
A. Bier, V. Sushchansky 21
We also deduce and recall the universal properties of the group P∞. In
Section 5 we discuss another representation of elements of P∞, proposed
by L. Kaluzhnin for p-elements of automorphism groups of either finite
or infinite p-adic rooted trees. Some basic results on this notation and its
properties are recalled here after [13]. We note that some generalization
of Kaluzhnin’s representation is proposed and investigated in [4].
Section 6 is devoted to subgroups in P∞ of a special type — the so
called ideal subgroups. We recall and prove a number of observations on
ideal subgroups of P∞ and use it in Sections 7 and 8 for discussion on
the lower central series and isometrically characteristic subgroups of P∞.
2. Rooted p-trees of finite height
and groups of their automorphisms
Let T (v) denote a rooted tree, i.e. a connected graph with no cycles
and a designated vertex v called the root. Every two vertices u and w
of the tree are connected with a unique path. The length of this path,
i.e. the number of edges in the path, is called the distance between u
and w. The set of all vertices of the tree T (v) can be partitioned into
subsets, called levels, with respect to the distance of the given vertex to
the root v. The n-th level, denoted by Ln is defined by the distance n, and
L0 = {v}. Given a vertex w ∈ Ln the number of vertices in Ln+1 which
are adjacent to w is called the (branch) index of w. A rooted tree is called
level-homogeneous if at every level the indices of all vertices are equal.
For brevity, we will call the common index of the vertices in level Ln —
the index of level Ln. It is clear that the index level of Ln is equal to
|Ln|/|Ln−1|. A level homogeneous tree can be uniquely characterized (up
to isomorphism) by a sequence (either finite or infinite) κ = (n1, n2, ..., )
of level indices; we denote this tree by Tκ. If there exists N , such that
LN 6= ∅ and LN+k = ∅ for all k > 0, then N is called the height of the
rooted tree.
In this paper we are concerned with the p-trees, which are defined
to be level-homogeneous rooted trees such that the indices related to
every level of the tree (except the last level) are p-powers (p — a prime).
Namely, given a sequence p = {pni}N
i=1 of natural powers of a prime p,
we define a rooted p-tree Tp(v,N) with root v as a level-homogeneous
tree in which the i-th level has the index pni , 1 < i < N and the last
level has index 0. One special example of a p-tree is the p-adic rooted
22 Kaluzhnin’s representations
tree Tp(v,N), in which all level indices, except those in the first and the
last level, are equal to p (i.e. the sequence p is constant).
A bijection τ from the set of vertices of T to itself is called an auto-
morphism of the tree T , if it preserves the adjacency of vertices. When
discussing automorphisms of a rooted tree, one additionally requires an
automorphism to preserve the root. The set of all automorphisms of the
tree T with the operation of morphism composition constitutes a group,
which we will denote by AutT . Let us first recall the well known result
on the structure of AutTp(v,N):
Proposition 2.1. Let p = {pni}N
i=1 be a sequence of natural powers of
prime p. Then
AutTp(v,N) ∼=
N
≀
i=1
Spni .
In particular, for a p-adic rooted tree Tp(v,N) of height N we have
AutTp(v,N) ∼=
N
≀
i=1
S(i)
p .
The proposition shows the mutual correspondence of groups of automor-
phisms of a p-adic rooted tree and certain transformation groups. This can
be easily seen by studying the action of a p-adic rooted tree automorphism
on the last level of the tree. Every such action is a permutation from SpN ,
where N is the height of the tree.
In his papers [8–11,13], L. Kaluzhnin studied the Sylow p-subgroups
of the symmetric group Spn . His investigations begin with the following
results:
Lemma 2.2. The Sylow p-subgroup Sylp(SpN ) of the symmetric group
SpN is isomorphic to a N-iterated wreath power of cyclic groups Cp of
degree p:
Sylp(SpN ) ∼=
N
≀
i=1
C(i)
p .
This result may be extended to the automorphism group of any rooted
p-tree:
Proposition 2.3. Let p = {pni}N
i=1 be a sequence of natural powers of
prime p. Then every Sylow p-subgroup Sylp(AutTp(v,N)) is isomorphic
to the M -iterated wreath power of cyclic groups of degree p:
Sylp(AutTp(v,N)) ∼=
M
≀
i=1
C(i)
p ,
where M = n1 + n2 + ...+ nN .
A. Bier, V. Sushchansky 23
Proof. It is enough to see that the iterated wreath power of permutation
groups is associative. Then we have:
Sylp(AutTp(v,N)) ∼=
N
≀
i=1
Sylp(Spni ) ∼=
N
≀
i=1
ni
≀
j=1
C(j)
p
∼=
M
≀
i=1
C(i)
p ,
where M stands for the sum of all ni, i = 1, ..., N .
The above proposition shows in particular, that the study of Sylow
p-subgroup of a rooted p-tree bears the same class of groups as Sylow
p-subgroups of a p-adic rooted tree. In the following, the Sylow p-subgroup
of AutTp(v, n) will be denoted by Pn.
3. Automorphism groups of infinite p-trees
So far we have discussed only finite trees, i.e. trees with finitely many
vertices. An infinite tree is said to be locally finite, if the branch index of
every vertex in that tree is finite. As particular examples of locally finite
trees one may consider the (infinite) rooted p-tree Tp(v) and an (infinite)
p-adic rooted tree Tp(v). These are trees with infinitely many levels (hence
no leaves) and infinitely many infinite paths starting at root (rays). The
uncountably infinite set ∂Tp(v) of all rays is called the boundary of the
tree. In the set ∂Tp(v) we introduce a metric ∆ in the following way. For
two rays α, β ∈ ∂T we set the distance as ∆(α, β) = 1
2k , where k is the
highest number of the level, such that both α and β contain a common
vertex in Lk. If such level does not exist, then the rays coincide and we
set ∆(α, β) = 0. Then (∂T,∆) is a compact ultrametric space.
Let AutTp(v) be the group of automorphisms of the infinite p-adic
rooted tree and let P∞ be its Sylow p-subgroup. We have:
Proposition 3.1. For every infinite sequence p = {pni}∞
i=1 of positive
powers of a prime p the Sylow p-subgroup of AutTp(v) is isomorphic to
P∞
∼=
∞
≀
i=1
C(i)
p .
Remark 3.2. We mention that if p = 2 then the Sylow 2-subgroup
of AutT2(v) coincides with AutT2(v) if and only if rhe sequence 2 is
constant, i.e. 2 = (2, 2, 2, ...).
We mention here that P∞ can be alternatively constructed as an
inverse limit of the inverse system (Pn, ψn) of Sylow p-subgroups in
24 Kaluzhnin’s representations
automorphism groups of finite trees Tp(v, n), with natural projections
ψn : Pn+1 →֒ Pn, n ∈ N (acting by neglecting the action on the last level
of the tree).
We now recall a useful and illustrative notation for elements of
AutTp(v) – the portrait of automorphisms of Tp(v). If f ∈ AutTp(v)
is an automorphism, then at every level Ln+1 of the tree f acts as a
permutation with distinct cycles on every set of descendants of vertices
in Ln. This is imposed by the requirement on automorphism to preserve
the adjacency of vertices. Hence, at the level Ln+1 one may illustrate the
action of f as shown in Fig. 1.
L
n
L
n+1
"
1 "
2
" np
Figure 1. A part of the portrait of an automorphism f ∈ AutTp(v).
Every automorphism f ∈ AutTp(v) acts in a natural way on the
boundary ∂Tp(v). Hence AutTp(v) acts on the compact ultrametric space
(∂T,∆). Therefore the group AutTp(v) is metrizable in a standard way.
Namely, for any two automorphisms f, g ∈ AutTp(v) we put
d(f, g) := max
α∈∂T
(αf , αg).
According to this definition the distance d(f, g) = 2−k, where k is the
number of the highest level in Tp(v), such that the portraits of automor-
phisms f and g are the same on the first k levels of Tp(v). If such a
number does not exist, i.e. the automorphisms f and g have identical
portraits, then we set d(f, g) = 0. The topology induced on AutTp(v) by
the ultrametric d is a standard profinite topology.
4. Residually finite groups
as automorphism groups of trees
Let G be a countably generated residually finite group and let Σ be a
series of subgroups of G:
G = G1 ⊇ G2 ⊇ G3 ⊇ ..., (1)
A. Bier, V. Sushchansky 25
such that
∞⋂
i=1
Gi = {1} and the factor groups Gi/Gi+1 are finite. We
construct the coset rooted tree T (Σ) in the following way. The root
corresponds to the whole group G, and for i > 1 the vertices of the i-th
level of T (Σ) are the right cosets G/Gi. A vertex Gix of i-th level is
connected with a vertex Gi+1y of the i + 1-st level (i > 0), if and only
if Gi+1y ⊂ Gix. The constructed rooted tree is level-homogeneous with
level indices equal to si = [Gi : Gi+1] (see Fig. 2). As the level indices are
finite, T (G) is a locally finite tree.
Gx
s
1
Gx
2
Gx
1
G
G y
1
1 1 1
2 G y
22
G ys2
2
G ys2
2 s1
Figure 2. Construction of a coset tree T (Σ).
It is well known that the function δ defined on G by the equation
δ(x, y) = 2−k ⇔ xy−1 ∈ Gk \Gk+1,
for every x, y ∈ G such that x 6= y and such that δ(x, x) = 0 is a metric
on G. It is easy to check that δ is an ultrametric on G. Namely, if x 6= y
and δ(x, y) = 2−k then, by definition, xy−1 ∈ Gk \ Gk+1 and hence
y−1x = (xy−1)−1 ∈ Gk \ Gk+1, i.e. δ(y, x) = 2−k as well. Moreover, if
δ(x, y) = 2−k and δ(y, z) = 2−j then we have
xy−1 ∈ Gk \Gk+1 and yz−1 ∈ Gj \Gj+1.
From this we get xz−1 = xy−1yz−1 ∈ Gk ∪Gj = Gmin{k,j}. Thus
δ(x, z) 6 2− min{k,j} = max{2−k, 2−j} = max{δ(x, y), δ(y, z)}
hence the group (G, δ) is an ultrametric space.
26 Kaluzhnin’s representations
Now, for every element g ∈ G we define the mapping ϕg : G −→ G with
the action ϕg : x 7−→ xg, x ∈ G. It is clear that for any subgroup H 6 G,
ϕg maps a right cosetH to the right cosetHg. Let us investigate the action
induced by ϕg on the tree T (Σ). Assume Gi = Gi+1 ∪Gi+1h1 ∪ .... be the
coset partition of Gi with respect to Gi+1. Then the coset Gigj is adjacent
to Gi+1hgj and hence for an arbitrary g ∈ G the images ϕg(Gigj) = Gigjg
and ϕg(Gi+1hgj) = Gi+1hgjg are adjacent as well. Thus ϕg induces an
automorphism ϕg of the coset tree T (Σ). The described correspondence
of elements of G and automorphisms of T (Σ) defines an embedding:
θ : G →֒ AutT (Σ), θ(g) = ϕg, g ∈ G.
Theorem 4.1. If the series Σ of subgroups G = G1 ⊇ G2 ⊇ G3 ⊇ ... is
normal, then the embedding θ : G →֒ AutT (Σ) is continuous and we have
G →֒
∞
≀
i=1
Gi/Gi+1.
Proof. (i) We show that embedding θ is a continuous map of one metric
space to the other. Assume that g0 ∈ G and 1
2k < ǫ < 1
2k−1
. Let g be an
element of G such that
δ(g, g0) 6 ε,
i.e. gg−1
0 ∈ Gk \Gk+1. Consider the action of θ(g0) and θ(g) on the tree
T (Σ). Since Gi is normal in G we have:
gg−1
o ∈ Gi ⇒ Gixgg
−1
o = Gix ⇒ Gixg = Gixg0,
i.e. θ(g) and θ(g0) act in the same way on the first k levels of T (Σ). It
follows that
d(θ(g0), θ(g)) 6
1
2k
< ε.
Thus the embedding θ is continuous.
(ii) If g ∈ G then θ(g) = ϕg is an automorphism of the tree T (Σ).
Hence we have an embedding
θ : G →֒ AutT (Σ) =
∞
≀
i=1
Sym(Gi/Gi+1.
In particular, a right coset Gi+1x ⊂ Gi is mapped onto the right coset
Gi+1xg ⊂ Gig. Thus we have
θ(G) ⊂
∞
≀
i=1
Gi/Gi+1
A. Bier, V. Sushchansky 27
and hence there exists an embedding
G →֒
∞
≀
i=1
Gi/Gi+1.
The above theorem is a generalization of a famous result of Kaluzhnin
and Krasner [16, 17]. It yields an important universal property of the
automorphism groups AutT of level-homogeneous trees :
Corollary 4.2.
(i) Every finitely generated residually finite group embeds in AutT for
some level-homogeneous tree T .
(ii) Every profinite group of countable weight embeds in AutT for some
level-homogeneous tree T .
In fact one can observe even more.
Corollary 4.3. Let κ = (n1, n2, ...) be an increasing sequence of indices.
Every finitely generated residually finite group and every profinite group
of countable weight embeds in AutTκ.
Proof. Let κ = (n1, n2, ...) be the sequence as required, and let G be a
finitely generated residually finite group with the descending series of
normal subgroups (1). Assume [Gi : Gi+1] = mi, i ∈ N. Then it is possible
to choose a subsequence κ′ = (ni1
, ni2
, ...) of κ such that for every j ∈ N
we have nij
> mj . Then, by Theorem 4.1, group G embeds in AutTκ′ ,
hence also in AutTκ.
If G is a p-group then one finds a sequence Σ of the form (1) with
cyclic factors of order p, i.e. [Gi : Gi+1] = p, i ∈ N. Groups possessing
such series of subgroups were called p∞ – groups by L. Kaluzhnin [12,15].
The coset tree T (Σ) in this case is a p-adic rooted tree and we have:
Corollary 4.4 ([12]). Every p∞-group is embeddable in P∞.
From this we have in particular
Corollary 4.5.
(i) P∞ is universal by embedding in the class of finitely generated
residually-p groups.
(ii) P∞ is universal by embedding in the class of pro-p groups of count-
able weight.
28 Kaluzhnin’s representations
Remark 4.6. The group P∞ is not the unique group with the universal
property described in Corollary 4.5. In [12] the author gives as the simplest
example the direct product G = P∞ × Cp of P∞ with a cyclic group of
order p. Then G has the universal property as it contains P∞. However,
it is not isomorphic to P∞ since G has a nontrivial center, while P∞ is
centerless.
5. Elements of P∞ as infinite sequences
of reduced polynomials
Following Kaluzhnin’s ideas in [13] we introduce a practical notation
for elements of the Sylow p-subgroup P∞. Namely, every u ∈ P∞ may be
represented as an infinite sequence (tableau) of reduced polynomials:
u = [a1, a2(x1), a3(x1, x2), ...],
where a1 ∈ Zp,
an(x1, x2, ..., xn−1)∈Zp[x1, x2, ..., xn−1]/〈xp
1−x1, ..., x
p
n−1−xn−1〉 for n>2.
Here 〈xp
1 − x1, ..., x
p
n−1 − xn−1〉 denotes the ideal of Zp[x1, x2, ..., xn−1]
generated by xp
i − xi, 1 6 i < n.
For example, the identity automorphism is represented by a table with
all coordinates equal to zero. If the first s − 1 coordinates in the table
u are zeros, and the s-th coordinate is nonzero, then u is said to have
depth s.
For every automorphism in P∞ defined by the portrait, one can find
its corresponding table representation applying the following procedure.
Let f ∈ P∞ be an automorphism of the tree Tp(v), given by its
portrait. First we label every vertex of the level Lk with a uniquely
assigned base p number with k digits in such a way that the ancestor
in level Lk−r of a given vertex v ∈ Lk is assigned a base p number
being the first k − r digits of the label of v. At given level, f acts on
descendants of a vertex by permutation. Hence the action of f on level
Lk+1 is given by a sequence of pk permutations, all of these being powers
of a fixed p-cycle α ∈ Sp. The portrait on level Lk+1 can be characterized
by the sequence (αs1 , αs2 , ..., α
s
pk ), or simply the sequence of exponents
(s1, s2, ..., spk), 0 6 si < p. Thus the action of f on the k + 1-st level of
Tp(v) is determined by a function, which assigns to every vertex vi in
Lk the respective exponent si. In particular, we define this function as
fk : Zp
pk
−→ Zp. It is known that every such function can be thought
A. Bier, V. Sushchansky 29
as a reduced polynomial over Zp. Then f is represented by the table
f = [f1, f2, ...].
For instance, we construct the tables of the generators of the (first)
Grigorchuk group [5]. The group is generated by the four automorphisms
of the binary tree T2: a, b, c and d with portraits presented in Fig. 3.
(a) Portrait of
automorphism a.
(b) Portrait of
automorphism b.
(c) Portrait of
automorphism c.
(d) Portrait of
automorphism d.
Figure 3. Portraits of generators of the first Grigorchuk group.
The tables that correspond to automorphisms of T2 consist of polyno-
mials over Z2 reduced by the ideals 〈x2
1 − x1, ..., x
2
n − xn〉. In particular,
the generators a, b, c and d can be presented as follows:
a = [1, 0, 0, 0, 0, 0, . . .],
b = [0, α2(x1), α3(x2), 0, α5(x4), α6(x5), 0, . . .],
c = [0, α2(x1), 0, α4(x3), α5(x4), 0, α7(x6), . . .],
d = [0, 0, α3(x2), α4(x3), 0, α6(x5), α7(x6), . . .],
where αn+1(xn) = αn+1(x1, x2, . . . , xn) = x1x2...xn−1(xn + 1).
For every n ∈ N the reduced polynomial an(x1, x2, ..., xn−1) is a sum
of monomials of the type m(x1, x2, ..., xn−1) = axǫ1
1 x
ǫ2
2 ...x
ǫn−1
n−1 , where
a ∈ Z
∗
p and 0 6 ǫi < p, i = 1, 2, ..., n − 1. Given a nonzero monomial
m(x1, x2, ..., xn−1) ∈ Zp[x1, ..., xn−1]/〈xp
1 − x1, ..., x
p
n−1 − xn−1〉 we define
its height h(m) to be equal
h(m) = 1 + ǫ1 + ǫ2 · p+ ...+ ǫn−1 · pn−2,
and we set h(0) = 0. Then for any reduced polynomial a(x1, x2, ..., xn−1) ∈
Zp[x1, ..., xn−1]/〈xp
1 − x1, ..., x
p
n−1 − xn−1〉 we define the height h(a) to be
the greatest height of all component monomials in a.
30 Kaluzhnin’s representations
The table notation of elements of P∞ is especially useful for com-
putations in the group. In particular, if u, v ∈ P∞ are automorphisms
represented by the tables
u = [a1, a2(x1), a3(x1, x2), ...], v = [b1, b2(x1), b3(x1, x2), ...],
then their product uv is given by the table uv = [c1, c2(x1), c3(x1, x2), ...]
with coordinates
cs(x1, ..., xs−1) = as(x1, x2, ..., xs−1) + bs ((x1, x2, ..., xs−1)u) , s > 1,
where (x1, x2, ..., xs1
)u = (x1−a1, x2−a2(x1), ..., xs−1−as−1(x1, ..., xs−2)).
In the following by ui, i > 1, we denote the beginning of length i of the
table u, i.e. ui = [a1, a2(x1), a3(x1, x2), ..., ai(x1, ..., xi−1)].
Below we list some results obtained by L. Kaluzhnin in [13] for the
group Pn of tables with n coordinates. The same arguments suffice to
show the analogous properties of infinite tables from P∞.
Lemma 5.1.
(i) For every reduced polynomial f(x1, ..., xs) and every table u ∈ Ps+1
the inequality holds:
h(f − fu) < h(f).
In particular, if h(f) = k there exists at least one element u ∈ Ps+1
such that the polynomial f(x1, ..., xs) − f((x1, ..., xs)
u) has height
equal to k − 1.
(ii) For every reduced polynomial f(x1, ..., xs) and every table u ∈ Ps+1
of depth r the following inequality holds:
h(f − fu) 6 ps − pr.
In particular, for every f there exists at least one element u ∈ Ps+1
of depth r such that the polynomial f(x1, ..., xs) − f((x1, ..., xs)
u)
has height equal to ps − pr.
(iii) Given an element u ∈ Ps+1 and a reduced polynomial u(x1, ..., xs) ∈
Zp[x1, ..., xs]/〈xp
1 − x1, ..., x
p
s − xs〉 the equation
f(x1, ..., xs) − f((x1, ..., xs)u) = g(x1, ..., xs)
has a solution f ∈ Zp[x1, ..., xs]/〈xp
1 − x1, ..., x
p
s − xs〉 and u ∈ Ps+1
if and only if the sum of values of g(x1, ..., xs) is zero in every orbit
of u.
A. Bier, V. Sushchansky 31
In the following [u]i, 1 6 1 < n, denotes the i-th coordinate of the
table u ∈ Pn.
Lemma 5.2. Let u and v be two elements of Ps+1 given by
u = [a1, a2(x1), ..., as+1(x1, x2, ...xs)]
v = [b1, b2(x1), ..., bs+1(x1, x2, ..., xs)].
(i) The conjugate uv = vuv−1 is represented by the table with coordi-
nates given as follows:
[uv]i = ai((x1, ..., xi−1)vi+
+bi(x1, ..., xi−1) − bi((x1, ..., xi−1)uv
i ).
(ii) For every table u ∈ Ps+1 of depth r there exists at least one table
v ∈ Ps+1 such that
h([vuv−1]i) > pi−1 − pr for every i > r + 1.
(iii) The commutator [u, v] = uvu−1v−1 is represented by the table with
coordinates given as follows:
[uvu−1v−1]i = ai((x1, ..., xi−1) − ai((x1, ..., xi−1)vu
i )+
+bi((x1, ..., xi−1)ui)+
−bi((x1, ..., xi−1)(uvu−1v−1)i).
(iv) The height of each coordinate of the commutator [u, v] satisfies the
inequality:
h([u, v]i) < min{h([u]i), h([v]i)}
or it is equal to 0. For every element u ∈ Ps+1 there exists at least
one element v ∈ Ps+1 such that
h([u, v]i) = h([u]i) − 1.
Lemmata 5.1 and 5.2 can be easily verified by direct calculations. For
detailed proofs of both lemmas we refer the reader to [13].
6. Ideal subgroups of P∞
Using the concept of height in the set of all sequences in P∞ we
introduce a partial order � on P∞ as follows. For any two elements
u = [a1, a2(x1), ...] and v = [b1, b2(x1), ...] from P∞ we set
u � v ⇔ h(ai) 6 h(bi) for all i ∈ N.
32 Kaluzhnin’s representations
One verifies directly that � is a partial order in P∞.
We recall that an ideal of a partially ordered set (X,6) is a nonempty
subset I ⊂ X satisfying the following two conditions:
(i) For every x ∈ I, if y 6 x then y ∈ I;
(ii) For every x, y ∈ I there exists z ∈ I, such that x 6 z and y 6 z.
Given an infinite sequence of nonnegative integers h = (hi)
∞
i=1, 0 6
hi < pi, in P∞ we consider the ideal subset
P (h) = {u ∈ P∞ | h(ai) 6 hi, i ∈ N}.
It is easy to see that P (h) is a subgroup of P∞, which we call the ideal
subgroup of P∞. We note that ideal subgroups were originally introduced
by L. Kaluzhnin in [13], who called them parallelotopic subgroups.
We present an alternative construction of the ideal subgroups. Let H
be an ideal subgroup, and let u be one of the maximal tables with respect
to � in H. By Hn we denote the subgroup of P∞, containing all tables
which have a unique nonzero coordinate in the n-th place and its height
does not exceed the height of [u]i:
Hn = {w ∈ P∞ | [w]i = 0 ∧ h([w]i) 6 h([u]i) for i 6= n}.
Then H =
∞∏
n=1
Hn, i.e. H is the closure of the direct product of subgroup
Hn, n ∈ N.
Theorem 6.1.
(i) Every ideal subgroup is closed in the profinite topology on P∞.
(ii) All ideal subgroups of P∞ constitute a distributive sublattice in the
lattice of all subgroups of P∞.
Proof. (i) Observe first that P∞ is a topological group with the topology
inherited from AutTp. The basis of neighborhoods of the identity in this
topology consists of subgroups Ds, s ∈ N, defined as follows:
Ds = {u ∈ P∞ | ui = 0 for i = 1, 2, ..., s}. (2)
All the subgroups Ds and their translations f · Ds, f ∈ P∞ are open.
Moreover, the subgroup Ds is the stabiliser of the s-th level of the tree Tp.
A. Bier, V. Sushchansky 33
Now, if P (h) is an ideal subgroup of P∞ defined by an infinite sequence
h = {hi}
∞
i=1, then
P (h) = P∞ \
∞⋃
s=1
⋃
f∗
s ∈P∞
f∗
s · Ds,
where f∗
s is a table from P∞ such that there exist a coordinate fi, 1 6 i 6 s
with h(fi) > hi. Thus P (h) is closed.
(ii) Let h = (hi)
∞
i=1 and h′ = (h′
i)
∞
i=1, 0 6 hi, h
′
i < pi, be two infinite
sequences of nonnegative integers, and let P (h) and P (h′) be the respective
ideal subgroups defined by these sequences. Observe that if h � h′ then
every element of P (h) is contained in P (h′) by the definition of the ideal.
Thus P (h) 6 P (h′). Moreover, in any case we have that h ∨ h′ = t
for t = {ti}
∞
i=1, where ti = max{hi, h
′
i}, i ∈ N; and h ∧ h′ = m for
m = {mi}
∞
i=1, where mi = min{hi, h
′
i}, i ∈ N. Thus the second statement
of the theorem follows.
The normality criterion obtained in [13] for ideal subgroups of the
finite groups Pm translates easily to the subgroups of P∞:
Theorem 6.2 ([14]). An ideal subgroup P (h) of depth r, defined by an
infinite sequence of nonnegative integers h = (hi)
∞
i=1 is normal in P∞ if
and only if for every i ∈ N the following inequality is satisfied:
hi > pi−1 − pr,
for all i > r + 1.
Proof. Let H = P (h) be a normal ideal subgroup of P∞ of depth r, and let
Ds, s ∈ N be the subgroups defined in (2). Direct calculations show that for
every s ∈ N the subgroup Ds is normal in P∞ and hence H · Ds is normal
in P∞ as well. Therefore H · Ds/Ds is normal in P∞/Ds
∼= Ps of depth r.
In [13] the author characterized all normal ideal subgroups of Ps as ideal
subgroups P (h1, h2, ..., hs) 6 Ps with hi > pi−1 − pr, where r is the depth
of the ideal subgroup. Therefore we have H · Ds/Ds
∼= P (h1, h2, ..., hs)
for every s ∈ N and thus H = P (h) with hi > pi−1 − pr, i ∈ N.
Conversely, assume H = P (h) to be an ideal subgroup of depth r
defined by a sequence h = (hi)
∞
i=1 satisfying hi > pi−1 − pr. Then clearly
H ·Ds/Ds
∼= P (h1, h2, ..., hs) is a normal subgroup of P∞/Ds, and therefore
H · Ds E P∞ for every s ∈ N. Thus the intersection
⋂
s∈N
H · Ds = H
is also normal in P∞.
34 Kaluzhnin’s representations
7. Lower central series of P∞
Knowing the form of commutators (see Lemma 5.2), it is possible to
characterize all terms of the lower central series in P∞. Namely
Theorem 7.1 ([14]). The k-th term γk(P∞) of the lower central series in
P∞ coincides with a normal ideal subgroup P (h) defined by the sequence
h = (hi)
∞
i=1, where hi = max{0, pi−1 − k}.
Proof. For k = 1 the description is valid. Assume then that the theorem
holds for all terms of the lower central series up till the k-th term. Consider
the k + 1-st term γk+1(P∞). By definition γk+1(P∞) is generated by the
commutators of the form [u, v], where u ∈ γk(P∞) and v ∈ P∞. Take
u = [0, 0, ..., 0, ai(x1, ..., xi−1), 0, 0, ...]. It follows from Lemma 5.2 and our
inductive assumption that
h([u, v]i) < h(u) 6 max{0, pi−1 − k},
hence h([u, v]i) 6 max{0, pi−1 −(k+1)}. Moreover, there always exists an
element v ∈ P∞ such that if only [u, v]i 6= 0 then h([u, v]i) = pi−1 −(k+1).
Therefore γk+1(P∞) contains every subgroup
Hi = {w ∈ P∞ | h([w]i) 6 pi−1 − (k + 1) ∧ [w]j = 0 for j 6= i},
and thus every product of elements of this type. It follows that γk+1(P∞) ⊇
∞∏
n=1
Hn and, since γk+1(P∞) is closed in P∞, we have
γk+1(P∞) =
∞∏
n=1
Hn = P (h),
where h = (hi)
∞
i=1, where hi = max{0, pi−1 − k}.
The above theorem is analogous to the respective characterization of
the lower central series in groups Pn, n ∈ N given in [13] by L. Kaluzhnin.
In particular, the derived subgroup P ′
∞ = γ2(P∞) = [P∞, P∞] is the ideal
subgroup P (h) defined by the sequence
h = (0, p− 2, p2 − 2, p3 − 2, . . . , pi−1 − 2, . . .).
From Theorem 7.1 we deduce a couple of observations.
Corollary 7.2. Every factor group γk(P∞)/γk+1(P∞) is a continual
elementary abelian group.
A. Bier, V. Sushchansky 35
Proof. One verifies directly, that every element of the factor group
γk(P∞)/γk+1(P∞)
is an infinite sequence (a1, a2, ...) where ai ∈ Zp for i ∈ N. Moreover, the
group operation for such sequences is simply the coordinate-wise addition.
Hence γk(P∞)/γk+1(P∞) ∼= Z
∞
p , as stated.
Corollary 7.3. P∞ is infinitely generated as a topological group.
Proof. Let us observe that P∞/P
′
∞ is a topological group with the quotient
topology induced from P∞. As P∞/P
′
∞
∼= Z
∞
p then P∞/P
′
∞ is not finitely
generated as a topological group. The statement of the corollary follows.
8. Isometrically characteristic subgroups of P∞
For the discussion on characteristic subgroups, we first recall some
known results on automorphism group of P∞. The groups of automor-
phisms of the groups Pn and P∞ were investigated in [3, 18–20]. In the
paper [20] the author characterizes all isometric automorphisms of the
group P∞, i.e. automorphisms preserving the basic sets Ds. In order to
formulate the main result of the recalled paper we first introduce the
notion of a scalar automorphism.
Let w̄ = {w1, w2, ...} be an infinite sequence of integers, wi ∈ Zp,
i ∈ N. The automorphism ω of P∞, defined by the rule:
ω([a1, a2(x1), ...]) = [w1a1, w2a2(x1w
−1
1 ), w3a3(x1w
−1
1 , x2w
−1
2 ), ...]
for any [a1, a2(x1), ...] ∈ P∞, is called a scalar automorphism. We denote
the set of all scalar automorphisms in P∞ by Ω∞. If p = 2 then Ω∞ is
trivial. The subgroup of all inner automorphisms of P∞ will be denoted
by InnP∞.
Lemma 8.1 ([20]). The subgroup Autis(P∞) of isometric automorphisms
of P∞ is decomposable into a general product of subgroups of inner and
scalar automorphisms:
Autis(P∞) = Ω∞ · InnP∞.
In this section we consider subgroups of P∞, which are invariant to
the isometric automorphisms. Subgroups of this kind were first discussed
by L. Kaluzhnin in the paper [14]. We recall here the results and include
a proof.
36 Kaluzhnin’s representations
Theorem 8.2 ([14]). Let p 6= 2. Then:
(i) Every normal ideal subgroup of P∞ is invariant to all isometric
automorphisms.
(ii) Every subgroup of P∞ which is invariant to all isometric automor-
phisms, is a normal ideal subgroup.
Proof. (i) Let H = P (h) to be a normal ideal subgroup of depth r of P∞.
From Theorem 6.2 it follows that H is defined by a sequence h = (hi)
∞
i=1
such that for every i > r we have hi > pi−1 − pr. It is enough to check
the invariancy of H with respect to scalar automorphisms form Ω∞.
Let ω ∈ Ω∞ be a scalar automorphism defined by a sequence of scalars
w̄ = {w1, w2, ...}, wi ∈ Zp, i ∈ N. Then, by definition, for every table
u = [a1, a2(x1), a3(x1, x2), ...] ∈ P∞ we have
ω([a1, a2(x1), ...]) = [w1a1, w2a2(x1w
−1
1 ), w3a3(x1w
−1
1 , x2w
−1
2 ), ...],
and hence h(ω(u)) = h(u). Thus ω(u) ∈ H, as stated.
(ii) Let H be a subgroup of P∞ which is invariant to all isometric
automorphisms and let α ∈ Autis P∞. Then, as Ds for s ∈ N are invariant
to isometric automorphisms, then α(H ·Ds) ⊂ H ·Ds. Moreover, α induces
an automorphism ᾱ of P∞/Ds
∼= Ps which by its isometric property, maps
table
u = [a1, a2(x1), a3(x1, x2), ..., as(x1, ..., xs)] ∈ H · Ds/Ds
to a table ᾱ(u) ∈ H ·Ds/Ds. Hence,H ·Ds/Ds is isomorphic to a subgroup
Hs of Ps invariant to ᾱ for every ᾱ induced by an isometric automorphism
of P∞. By the results of L. Kaluznin from [13] we deduce that Hs is a
normal ideal subgroup of Ps. It follows that both subgroups H · Ds and
H are normal ideal subgroups of P∞.
From the above theorem and Theorem 6.1 it follows directly:
Corollary 8.3. If p 6= 2 then every subgroup of P∞ which is invariant
to all isometric automorphisms is closed.
Whether there exist characteristic subgroups of P∞ which are not
closed remains an interesting question to investigate, posed by Kaluzhnin
in [14].
A. Bier, V. Sushchansky 37
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Contact information
A. Bier,
V. Sushchansky
Institute of Mathematics
Silesian University of Technology
ul. Kaszubska 23, 44-100 Gliwice, Poland
E-Mail(s): agnieszka.bier@polsl.pl,
vitaliy.sushchanskyy@polsl.pl
Received by the editors: 09.03.2015
and in final form 09.03.2015.
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