Fully invariant subgroups of an infinitely iterated wreath product

Abstract

The article deals with the infinitely iterated wreath product of cyclic groups C p of prime order p. We consider a generalized infinite wreath product as a direct limit of a sequence of finite nth wreath powers of C p with certain embeddings and use its tableau representation. The main result are the statements that this group doesn't contain a nontrivial proper fully invariant subgroups and doesn't satisfy the normalizer condition.

Full text

Journal Algebra Discrete Math.
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 12 (2011). Number 2. pp. 85 – 93
c
Journal “Algebra and Discrete Mathematics”
Fully invariant subgroups
of an infinitely iterated wreath product
Yuriy Yu. Leshchenko
Communicated by V. I. Sushchansky
A b s t r a c t . The article deals with the infinitely iterated
wreath product of cyclic groups
Cp
of prime order
p
. We consider a
generalized infinite wreath product as a direct limit of a sequence of
finite
n
th wreath powers of
Cp
with certain embeddings and use its
tableau representation. The main result are the statements that this
group doesn’t contain a nontrivial proper fully invariant subgroups
and doesn’t satisfy the normalizer condition.
Introduction
Awreath product of permutation groups is a group-theoretical con-
struction, which is widely used for building groups with certain special
properties. Given two permutation groups (G1, X1)and (G2, X2), where
G1
acts on
X1
and
G2
acts on
X2
, we denote their wreath product to be
the permutation group
G1≀G2
=
{
[
g1
(
x
)
, g2
]
|g2∈G2, g1
:
X2→G1},
which acts on the direct product
X1×X2
(imprimitive action). The no-
tion of the wreath product of two groups can be easily generalized to an
arbitrary finite number of factors. If all factors are isomorphic to
G
then
the corresponding wreath product is often called the wreath power of
G
(or the n-iterated wreath product of G).
Given a residue field
Zp
(
p
is prime) we consider its additive group
Cp
(without loss of generality we can assume that
Cp
acts on itself by the
right translations). The finite
n
th wreath power of
Cp
(which is isomorphic
to the Sylow
p
-subgroup of the finite symmetric group
Spn
) was studied
2000 Mathematics Subject Classification: 20B22, 20E18, 20E22.
Key words and phrases: wreath product, fully invariant subgroups.

Journal Algebra Discrete Math.
86 F u l ly i n va r i a n t su b g r o u p s o f wr e a t h p o w e r
by L. A. Kaloujnine [
7
]. In particular, in [
7
] the author investigated the
structure of finite wreath powers of cyclic permutation groups (charac-
teristic subgroups, upper and lower central series and derived series were
described). Later, similar results for wreath powers of elementary abelian
groups were obtained by V. I. Sushchansky in [15].
The notion of iterated wreath product admits various generalizations
in the case of an infinite number of factors. In [
4
] P. Hall introduced a
general construction
W=wrλ∈ΛGλ
of the ”restricted” (in the sense of an action as a permutation group)
wreath product of permutation groups indexed by a totally ordered set.
In the same article the author used this wreath product construction to
obtain the examples of characteristically simple groups.
Similar approaches can be found in papers of I. D. Ivanuta [
6
], W. C.
Holland [5] (unrestricted wreath product of permutation groups indexed
by a partially ordered set), M. Dixon and T. A Fournelle ([1] and [2]).
For example, in [
6
] the author adapted the Hall’s general construc-
tion with factors indexed by a totally ordered set to describe the main
(transitive) infinite Sylow
p
-subgroup of the finitary symmetric group. To
operate with considered wreath product I. D. Ivanuta also used a tableau
representation of its elements.
The normalizer condition for the direct limits of finite standard wreath
products (so called Kargapolov groups) was studied in [
13
] by Yu. I.
Merzlyakov. Also in [
14
] a criterion of self normalizability for some classes
of subgroups of the finitary unitriangular group was established.
In this article we consider the infinite wreath product construction
(denoted by
Uω
p
) as a direct limit of a sequence of finite
n
th wreath
powers of
Cp
with certain embeddings. We also use the so called tableau
representation of Uω
pfor the study of its properties.
In the first section the generalized infinite wreath product
Uω
p
is
defined. Then, in section 2 we present a review of known results on the
characteristic structure of
Uω
p
. The main results of the article are given in
the last section:
1)
if
p6
= 2 and
R
is a fully invariant subgroup of
Uω
p
then either
R=E
(the identity subgroup), or
R
=
Uω
p
; in other words,
Uω
p
is fully
invariantly simple (theorem 2);
2) Uω
pdoesn’t satisfy the normalizer condition (theorem 3).
The author wishes to thank Professor V. I. Sushchansky for his advice in
the preparation of this paper.

Journal Algebra Discrete Math.
Y u . L e s h c h e n k o 87
1. Generalized wreath product
In this section we consider a group of infinite tableaux of reduced
polynomials (an approach similar to what was proposed by L.A. Kaloujnine
in [
7
]) and then define it as a direct limit of finitely iterated wreath products
(generalized wreath product).
1.1. The tableau representation
Let
p
be a prime (
p6
= 2) and
Cp
be the additive group of the residue
field
Zp
. In other words,
Cp
is the cyclic additive group of order
p
, which
acts on itself by the right translations. Define
Uω
p
as a group of infinite
almost zero tableaux
[a1(x2, . . . , xk), . . . , an(xn+1, . . . , xk),0, . . .], k, n ∈N, k > n, (1)
where
ai
(
xi+1, . . . , xk
)is a polynomial over
Zp
reduced (degree of each
variable ≤p−1) modulo the ideal
hxp
i+1 −xi+1, xp
i+2 −xi+2, . . . , xp
k−xki.
The group Uω
pacts on the direct product
X=
∞
Y
i=1
Zp={(t1, . . . , tm,0, . . .)|ti∈Zp, m ∈N}(2)
(
X
is the set of all almost zero sequences over
Zp
). If
u∈Uω
p
and
t
=
(ti)∞
i=1 ∈Xthen
tu= (t1+a1(t2, . . . , tk), . . . , tn+an(tn+1, . . . , tk), tk+1 , tk+2, . . .).(3)
For simplicity, we introduce some auxiliary notation. Let
ai(xi+1,k) = ai(xi+1 , . . . , xk)
and [
u
]
i
=
ai
(
xi+1,k
)– the
i
th coordinate of the tableau
u
. Also, let
[
ai
(
xi+1,k
)]
∞
i=1
be a short notation of the tableau (1) and
f
(
xu
)denote the
reduced polynomial, which is equivalent to the polynomial
f(...,xj+aj(xj+1, . . . , xk), . . .).
Thus, according to (3), if
u
= [
ai
(
xi+1,k
)]
∞
i=1
and
v
= [
bi
(
xi+1,k
)]
∞
i=1
then
uv = [ai(xi+1,k) + bi(xu
i+1,k)].(4)
If [
u
]
n6
= 0 and [
u
]
i
= 0 for all
i > n
then
n
is called the depth of the
tableau u.

Journal Algebra Discrete Math.
88 F u l ly i n va r i a n t s u b g r o u p s o f w r e a t h p o w e r
1.2. Uω
pas a direct limit of wreath powers
Recall that a sequence
{Gn}∞
n=1
of groups with a corresponding se-
quence
{ϕn:Gn→Gn+1}∞
n=1
of embeddings is called direct system and denoted by hGn, ϕni∞
n=1.
Let
Pn
be a Sylow
p
-subgroup of the symmetric group
Spn
(
p
is prime
and
n∈N
). In [
7
] the group
Pn
was described by L.A. Kaloujnine as a
group of tableaux
[a1, a2(x1), . . . , an(x1, x2, . . . , xn−1)],(5)
where
a1∈Zp
,
ai
(
x1, . . . , xi−1
)is a polynomial (over the residue field
Zp
)
reduced modulo the ideal
hxp
1−x1, xp
2−x2, . . . , xp
i−1−xi−1i.
If we denote by
ai
(
x1, . . . , xi−1
)and
bi
(
x1, . . . , xi−1
)the
i
th coordinates
of tableaux
u
and
v
respectively then the
i
th coordinate of
u·v
can be
found as follows
ai(x1, . . . , xi−1) + bi(x1+a1, . . . , xi−1+ai−1(x1, . . . , xi−2)).
Also
Pn
can be considered as the
n
th wreath power of a cyclic group
Cp
,
i.e. Pn=Cp≀. . . ≀Cp(nfactors).
Given
Pn
and
Pn+1
(
n∈N
) define the mapping
δn
:
Pn→Pn+1
. If
u
is a tableau of type (5) then
δn(u) = [0, a1, a2(x2), . . . , an(x2, . . . , xn)] ∈Pn+1.
By the direct calculations (or see [
9
], Lemma 4) it is easy to show that
δn
is
astrictly diagonal (in the sense of the article [
8
]) embedding. Moreover,
δn
is, actually, the embedding of
Pn
onto the diagonal of the wreath product
Pn+1 =Pn≀Cp, where Cpis the active group.
Lemma 1.
[
9
]The group
Uω
p
is isomorphic to the direct limit of the direct
system hPn, δni∞
n=1.
2. Characteristic subgroups of Uω
p
This section is devoted to some necessary statements regarding de-
scription of characteristic subgroups of
Uω
p
. All results are taken from [
10
]
and [
11
], where they are proved for the case of a generalized infinitely
iterated wreath product of elementary abelian groups.

Journal Algebra Discrete Math.
Y u . L e s h c h e n k o 89
Definition 1.
The weighted degree of the monomial
xk1
1xk2
2. . . xkn
n
is the
positive rational number
h=
n
X
i=1
kip−i+ 1.
The weighted degree of a polynomial is the maximum among the weighted
degrees of its monomials. Let also h[0] = 0.
Thus, if
u
= [
ai
(
xi+1,k
)]
∞
i=1 ∈Uω
p
then
h
[
ai
(
xi+1,k
)]
∈ {
0
} ∪
[1; 1 +
p−i
).
Given
u
= [
ai
(
xi+1,k
)]
∞
i=1 ∈Uω
p
we denote the weighted degree of
ai
(
xi+1,k
)by
|u|i
. The sequence
|u|
= (
|u|i
)
∞
i=1
is called the multidegree
of the tableau
u
. The set of all multidegrees can be partially ordered as
follows:
|u|  |v|
if and only if
|u|i≤ |v|i
(with respect to the natural
order on Q) for all i∈N.
Definition 2.
A subgroup
R
of the group
Uω
p
is called a parallelotopic
subgroup if u∈Rand |v|  |u|yield v∈R.
For every parallelotopic subgroup
R
we put in correspondence the
sequence |R|= (χε
i)∞
i=1 such that
1) χi= supu∈R|u|i;
2) if Rcontains such a tableau uthat |u|i=χi, then ε=”+”;
3) otherwise, ε=”−”.
This sequence is called the indicatrix of
R
. If
χi6
= 0 for finitely many
indices
i
only then
d
(
R
) =
max{i|χi6
= 0
}
is called the depth of the
parallelotopic subgroup R. Otherwise, we put d(R) = +∞.
Definition 3.
A group
G
is called characteristically (fully invariantly
or verbally)simple if only its characteristic (fully invariant or verbal)
subgroups are E(the identity subgroup) or G.
In [
12
] it was shown that
Uω
p
is verbally complete (a group
G
is
called verbally complete if for an arbitrary
g∈G
and for an arbitrary
non-trivial word
w
(
x1, x2, . . . , xn
)there are
g1, g2, . . . , gn∈G
such that
w(g1, g2, . . . , gn) = g). Consequently, Uω
pis verbally simple.
Characteristic subgroups of
Uω
p
were investigated in [
10
] and [
11
].
Moreover, in these papers even more general case (the infinitely iterated
wreath powers (we denote it by
U∞
p,n
) of the elementary abelian groups
of rank
n
) was considered. If we put
n= 1
then
U∞
p,1∼
=Uω
p
. Thereby,
from [
10
] and [
11
] it is known that
Uω
p
has non-trivial proper characteristic
subgroups, i.e. Uω
pis not characteristically simple.

Journal Algebra Discrete Math.
90 F u l ly i n va r i a n t s u b g r o u p s o f w r e a t h p o w e r
Theorem 1.
[
11
]If
p6
= 2 and
R
is a characteristic (fully invariant or
verbal)subgroup of the group Uω
p, then
1) Ris a parallelotopic subgroup of Uω
p;
2) d(R)<+∞(Rhas finite depth).
3. Fully invariant subgroups of Uω
p
Recall that a subgroup
H
of a group
G
is called fully invariant if it is
invariant under all endomorphisms (homomorphisms of Ginto G) of G.
Lemma 2. Let u= [ai(xi+1,k )]∞
i=1 ∈Uω
p. Then the mapping
ϕ:Uω
p→Uω
p,
which acts on the elements of Uω
pby the rule:
[ϕ(u)]1= 0,[ϕ(u)]i+1 =ai(xi+2, . . . , xk+1 )fol all i∈N
is an endomorphism of Uω
p.
Remark 1.
For any
u∈Uω
p
the mapping
ϕ
actually is a coordinate-wise
translation to the right (all variables also must be shifted: xj→xj+1).
Proof of lemma 2.Let
u= [ai(xi+1,k)]∞
i=1 ∈Uω
pand v= [bi(xi+1,k)]∞
i=1 ∈Uω
p
(without loss of generality we assume that [
u
]
i
= [
v
]
i
= 0 for all
i > n
, where
n < k
). Obviously, [
ϕ
(
uv
)]
1
= [
ϕ
(
u
)
ϕ
(
v
)]
1
= 0. Therefore, we consider
[
ϕ
(
uv
)]
i
and [
ϕ
(
u
)
ϕ
(
v
)]
i
, where
i≥
2. According to the formula (4) we
have
[uv]i=ai(xi+1,k ) + bi(xu
i+1,k) =
=ai(...,xj, . . .) + bi(...,xj+aj(xj+1, . . . , xk), . . .),
where j∈ {i+ 1, . . . , k}. Thus
[ϕ(uv)]i+1 =ai(...,xj+1 , . . .) + bi(...,xj+1 +aj(xj+2 , . . . , xk+1 ), . . .),
where j∈ {i+ 1, . . . , k}.
On the other hand, since
[ϕ(u)]i+1 =ai(xi+2, . . . , xk+1 )and [ϕ(v)]i+1 =bi(xi+2, . . . , xk+1 )
then
[ϕ(u)ϕ(v)]i+1=ai(...,xj+1, . . .)+bi(...,xj+1 +[ϕ(u)]j+1, . . .) =
=ai(...,xj+1, . . .)+bi(...,xj+1 +aj(xj+2 , . . . , xk+1 ), . . .),
where j∈ {i+ 1, . . . , k}.
Hence, ϕ(uv) = ϕ(u)ϕ(v), i.e. ϕis an endomorphism of Uω
p.

Journal Algebra Discrete Math.
Y u . L e s h c h e n k o 91
Now we can prove the main result.
Theorem 2. If p6= 2 then Uω
pis fully invariantly simple.
Proof.
Let us assume that
R
is a fully invariant subgroup of the group
Uω
p
,
R6
=
E
and
R6
=
Uω
p
. Then
R
is a parallelotopic subgroup of
Uω
p
and
d(R)<+∞(theorem 1). If d(R) = rthen Rcontains the tableau
u= [0,...,0
|{z }
r−1
,1,0, . . .].
Given the endomorphism
ϕ
, defined in lemma 2, we have
ϕ
(
u
)
6∈ R
(since
the depth of
ϕ
(
u
)is equal to
r
+ 1). But, on the other hand,
ϕ
(
u
)
∈R
(since
R
is a fully invariant subgroup of
Uω
p
). This contradiction shows
the falsity of the assumptions.
4. A note on the normalizer condition
A group
G
is said to satisfy the normalizer condition if every proper
subgroup
H
is properly contained in its own normalizer, i.e.
H < NG(H)
for all H < G.
Lemma 3.
[
13
]If a group
G
satisfies the normalizer condition then every
subgroup Hof Galso satisfies normalizer condition.
Let
δij =1, if i=j;
0, if i6=j.
Given a prime
p
we consider the set
FMω
p
of all almost identity infinite
matrices over Zp(ωis the least infinite ordinal). In other words,
FMω
p=(aij )
aij ∈Zp
;
aij
=
δij
for all but
finitely many (i, j)∈N×N.
Also,
FMω
p
is called the set of all finitary matrices. Finitary matrices can
be multiplied by the usual rule:
(ab)ij =X
k
aikbk j ,
since the sum on the right side contains only a finite number of nonzero
terms. The set of all finitary invertible matrices with operation of matrix
multiplication forms a group, which is called the finitary linear group.
The finitary (upper) unitriangular group is the group
U T ω
p
of all finitary
matrices over
Zp
such that
aij
=
δij
for all
i≥j
(see, for example, [
14
]).
Similarly, we can consider the lower unitriangular group.

Journal Algebra Discrete Math.
92 F u l ly i n va r i a n t s u b g r o u p s o f w r e a t h p o w e r
On the other hand, the group of all finitary invertible matrices can
be considered as a permutation group on the set
X
of all almost zero
sequences over Zp(see formula (2)) with natural action
tA=t·A= (
k
X
i=1
ai1ti,
k
X
i=1
ai2ti,...,
k
X
i=1
ainti, . . .),(6)
where t= (ti)∞
i=1 ∈Xand A= (aij )∈FMω
p,i, j ∈N.
In [
14
] (see theorem 1) it was shown that the finitary unitriangular
group
U T ω
p
does not satisfy the normalizer condition. And we get the
following natural corollary.
Theorem 3. The group Uω
pdoes not satisfy the normalizer condition.
Proof.
According to lemma 3it is sufficient to show that
Uω
p
contains a
group, which is isomorphic to the finitary (upper) unitriangular group.
Let
u∈Uω
p
be a tableau with linear coordinates, i.e. [
u
]
i
is a homoge-
neous linear polynomial. By the direct computation it can be shown that
the action of
u
on
X
(which is defined by the equation (3)) agrees with the
action (6). Obviously, the subset of all tableaux with linear coordinates is
a subgroup of
Uω
p
and this subgroup is isomorphic to the finitary (lower)
unitriangular matrix group.
Since the groups of upper and lower finitary unitriangular matrices
are isomorphic (with isomorphism
x→
(
xT
)
−1
, where
x∈U T ω
p
and
xT
is the transpose of
x
) the group
Uω
p
does not satisfy the normalizer
condition.
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Journal Algebra Discrete Math.
Y u . L e s h c h e n k o 93
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C o n ta c t i n f o r m a t i o n
Yu. Leshchenko
Department of Algebra and Mathematical Anal-
ysis, Bogdan Khmelnitsky National University,
81, Shevchenko blvd., Cherkasy, 18031, Ukraine
E-Mail: ylesch@ua.fm
Received by the editors: 15.04.2011
and in final form 19.12.2011.