Fully invariant subgroups of an infinitely iterated wreath product
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 nth wreath powers of Cp with certain embeddings and use its tableau representation. The main result are the...
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| Цитувати: | Fully invariant subgroups of an infinitely iterated wreath product / Y.L. Leshchenko // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 85–93. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1548422019-06-17T01:30:51Z Fully invariant subgroups of an infinitely iterated wreath product Leshchenko, Y.L. 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 nth 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. 2011 Article Fully invariant subgroups of an infinitely iterated wreath product / Y.L. Leshchenko // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 85–93. — Бібліогр.: 15 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20B22, 20E18, 20E22. http://dspace.nbuv.gov.ua/handle/123456789/154842 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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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 nth 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. |
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Leshchenko, Y.L. |
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Leshchenko, Y.L. Fully invariant subgroups of an infinitely iterated wreath product Algebra and Discrete Mathematics |
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Leshchenko, Y.L. |
| author_sort |
Leshchenko, Y.L. |
| title |
Fully invariant subgroups of an infinitely iterated wreath product |
| title_short |
Fully invariant subgroups of an infinitely iterated wreath product |
| title_full |
Fully invariant subgroups of an infinitely iterated wreath product |
| title_fullStr |
Fully invariant subgroups of an infinitely iterated wreath product |
| title_full_unstemmed |
Fully invariant subgroups of an infinitely iterated wreath product |
| title_sort |
fully invariant subgroups of an infinitely iterated wreath product |
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Інститут прикладної математики і механіки НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/154842 |
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Fully invariant subgroups of an infinitely iterated wreath product / Y.L. Leshchenko // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 2. — С. 85–93. — Бібліогр.: 15 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT leshchenkoyl fullyinvariantsubgroupsofaninfinitelyiteratedwreathproduct |
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2025-07-14T06:55:04Z |
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2025-07-14T06:55:04Z |
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1837604384490389504 |
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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
Abstract. 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 nth 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
A wreath 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 nth 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.
86 Fully invariant subgroups of wreath power
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 nth wreath
powers of Cp with certain embeddings. We also use the so called tableau
representation of Uω
p for 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 p 6= 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
p doesn’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.
Yu. Leshchenko 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 (p 6= 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
〈xpi+1 − xi+1, x
p
i+2 − xi+2, . . . , x
p
k − xk〉.
The group Uω
p acts on the direct product
X =
∞∏
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 ∈ X then
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 ith 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(x
u
i+1,k)]. (4)
If [u]n 6= 0 and [u]i = 0 for all i > n then n is called the depth of the
tableau u.
88 Fully invariant subgroups of wreath power
1.2. Uω
p as 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 〈Gn, ϕn〉
∞
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
〈xp1 − x1, x
p
2 − x2, . . . , x
p
i−1 − xi−1〉.
If we denote by ai(x1, . . . , xi−1) and bi(x1, . . . , xi−1) the ith coordinates
of tableaux u and v respectively then the ith 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 nth wreath power of a cyclic group Cp,
i.e. Pn = Cp ≀ . . . ≀ Cp (n factors).
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
a strictly 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 Cp is the active group.
Lemma 1. [9] The group Uω
p is isomorphic to the direct limit of the direct
system 〈Pn, δn〉
∞
n=1.
2. Characteristic subgroups of U
ω
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.
Yu. Leshchenko 89
Definition 1. The weighted degree of the monomial xk11 xk22 . . . xknn is the
positive rational number
h =
n∑
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 ∈ R and |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 R contains such a tableau u that |u|i = χi, then ε = ”+”;
3) otherwise, ε = ”−”.
This sequence is called the indicatrix of R. If χi 6= 0 for finitely many
indices i only then d(R) = max{i | χi 6= 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
p is 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
p is not characteristically simple.
90 Fully invariant subgroups of wreath power
Theorem 1. [11] If p 6= 2 and R is a characteristic (fully invariant or
verbal) subgroup of the group Uω
p , then
1) R is a parallelotopic subgroup of Uω
p ;
2) d(R) < +∞ (R has finite depth).
3. Fully invariant subgroups of U
ω
p
Recall that a subgroup H of a group G is called fully invariant if it is
invariant under all endomorphisms (homomorphisms of G into 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
p by 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
p and 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(x
u
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. ϕ is an endomorphism of Uω
p .
Yu. Leshchenko 91
Now we can prove the main result.
Theorem 2. If p 6= 2 then Uω
p is fully invariantly simple.
Proof. Let us assume that R is a fully invariant subgroup of the group
Uω
p , R 6= E and R 6= Uω
p . Then R is a parallelotopic subgroup of Uω
p and
d(R) < +∞ (theorem 1). If d(R) = r then R contains the tableau
u = [0, . . . , 0
︸ ︷︷ ︸
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 H of G also satisfies normalizer condition.
Let
δij =
{
1, if i = j;
0, if i 6= j.
Given a prime p we consider the set FMω
p of all almost identity infinite
matrices over Zp (ω is 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 =
∑
k
aikbkj ,
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 UTω
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.
92 Fully invariant subgroups of wreath power
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∑
i=1
ai1ti,
k∑
i=1
ai2ti, . . . ,
k∑
i=1
ainti, . . .), (6)
where t = (ti)
∞
i=1 ∈ X and A = (aij) ∈ FMω
p , i, j ∈ N.
In [14] (see theorem 1) it was shown that the finitary unitriangular
group UTω
p does not satisfy the normalizer condition. And we get the
following natural corollary.
Theorem 3. The group Uω
p does not satisfy the normalizer condition.
Proof. According to lemma 3 it 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 ∈ UTω
p and
xT is the transpose of x) the group Uω
p does not satisfy the normalizer
condition.
References
[1] M. Dixon, T. A. Fournelle, Some properties of generalized wreath products, Compo-
sitio Math., 52, N. 3 (1984), 355-372.
[2] M. Dixon, T. A. Fournelle, Wreath products indexed by partially ordered sets, Rocky
Mt. J. Math., 16, N. 1, (1986), 7-15.
[3] P. Hall, Some constructions for locally finite groups, J. London Math. Soc., 34
(1959),305-319.
[4] P. Hall, Wreath powers and characteristically simple groups, Proc. Cambridge Phil.
Soc., 58 (1962), 170-184.
[5] W. C. Holland, The characterization of generalized wreath products, J. Algebra,13
(1969), 152-172.
[6] I. D. Ivanuta, Sylow p-subgroups of the finitary symmetric group, Ukrain. Mat. Zh.,
15 (1963), 240-249.
[7] L. Kaloujnine, La structure des p-groupes de Sylow des groupes symetriques finis,
Ann. Sci. l’Ecole Norm. Super., 65 (1948), 239-276.
Yu. Leshchenko 93
[8] N. V. Kroshko, V. I. Sushchansky, Direct limits of symmetric and alternating groups
with strictly diagonal embeddings, Arch. Math. (Basel), 71 (1998), 173-182.
[9] Yu. Yu. Leshchenko, V. I. Sushchansky, Sylow structure of homogeneous symmetric
groups of superdegree p
∞, Mat. Stud., 22 (2004), 141-151.
[10] Yu. Yu. Leshchenko, Characteristic subgroups of the infinitely iterated wreath prod-
uct of elementary abelian groups, Ukr. Math. Bull., 6, N. 1 (2009), 37-50.
[11] Yu. Yu. Leshchenko, Infinitely iterated wreath power of elementary abelian groups,
Mat. Stud., 31 (2009), 12-18.
[12] Yu. Yu. Leshchenko, The structure of one infinite wreath power construction of
the regular group of the prime order p, Mat. Stud., 28 (2007), 141-146.
[13] Yu. I. Merzlyakov, About Kargapolov groups, Dokl. Akad. Nauk SSSR, 322 (1992),
41-44.
[14] Yu. I. Merzlyakov, Equisubgroups of unitriangular groups: a criterion of self nor-
malizability, Dokl. Russian Akad. Nauk, 50 (1995), 507-511.
[15] V. I. Sushchansky, A wreath product of elementary abelian groups, Math. Notes,
11 (1972), 61-72.
Contact information
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.
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