A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees”
The results on automorphisms of homogeneous alternating groups are corrected and improved.
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| Дата: | 2005 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут прикладної математики і механіки НАН України
2005
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/156612 |
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| Цитувати: | A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” / Y.V. Lavrenyuk, V.I. Sushchansky // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 90–92. — Бібліогр.: 2 назв. — англ. |
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irk-123456789-1566122019-06-19T01:29:07Z A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” Lavrenyuk, Y.V. Sushchansky, V.I. The results on automorphisms of homogeneous alternating groups are corrected and improved. 2005 Article A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” / Y.V. Lavrenyuk, V.I. Sushchansky // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 90–92. — Бібліогр.: 2 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20B35, 20E08, 20F28, 20F50. http://dspace.nbuv.gov.ua/handle/123456789/156612 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
The results on automorphisms of homogeneous
alternating groups are corrected and improved. |
| format |
Article |
| author |
Lavrenyuk, Y.V. Sushchansky, V.I. |
| spellingShingle |
Lavrenyuk, Y.V. Sushchansky, V.I. A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” Algebra and Discrete Mathematics |
| author_facet |
Lavrenyuk, Y.V. Sushchansky, V.I. |
| author_sort |
Lavrenyuk, Y.V. |
| title |
A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” |
| title_short |
A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” |
| title_full |
A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” |
| title_fullStr |
A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” |
| title_full_unstemmed |
A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” |
| title_sort |
note to our paper “automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/156612 |
| citation_txt |
A note to our paper “Automorphisms of homogeneous symmetric groups and hierarchomorphisms of rooted trees” / Y.V. Lavrenyuk, V.I. Sushchansky // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 90–92. — Бібліогр.: 2 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
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| first_indexed |
2025-07-14T09:00:18Z |
| last_indexed |
2025-07-14T09:00:18Z |
| _version_ |
1837612263291224064 |
| fulltext |
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Algebra and Discrete Mathematics A SHORT NOTE
Number 2. (2005). pp. 90 – 92
c© Journal “Algebra and Discrete Mathematics”
A note to our paper “Automorphisms of
homogeneous symmetric groups and
hierarchomorphisms of rooted trees”
Yaroslav V. Lavrenyuk, Vitaly I. Sushchansky
Abstract. The results on automorphisms of homogeneous
alternating groups are corrected and improved.
This note is a supplement to the recent paper [LS]. We assume that
the paper is available to the reader, and freely use notations and termi-
nology introduced there.
It was falsely asserted in [LS] that every automorphism of A(∂TΘ) is
locally inner. So, some parts of assertions of Proposition 10, Corollary 2,
Theorem 13, namely the assertions in parenthesis, are false.
We give below slightly modified and corrected Proposition 10 with
corrected and more clear proof and state result on the automorphisms of
the group A(∂TΘ).
Proposition 1. Let X denote either H or AH. Let α ∈ AutXΩ be such
that α(Xn) ≤ Xk where 1 < n ≤ k. Then α|Xn
∈ InnHk.
Proof. Let g ∈ XΩ. We have g ∈ Xn for some n ∈ N. Since XΩ is union
of its subgroups Xn (n ∈ N), there exists k ∈ N such that
α(Xn) ≤ Xk.
Let us show that α|Xn
is induced by an inner automorphism of Hk. By
Corollary 3.13c of [Rub] the automorphism α is induced by a homeomor-
phism γ of ∂TΩ. Note that every homeomorphism belonging to Hk is de-
termined by its action on Vk(TΩ). Suppose that for some 1 ≤ i, j ≤ fΩ(n),
i 6= j and 1 ≤ l ≤ fΩ(k) we have
γ−1(Pkl) ∩ Pni 6= ∅, (1)
2000 Mathematics Subject Classification: 20B35, 20E08, 20F28, 20F50.
Key words and phrases: rooted tree, hierarchomorphism, local isometry, diag-
onal embedding, direct limit, homogeneous symmetric group, group automorphisms.
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.Y. V. Lavrenyuk, V. I. Sushchansky 91
γ−1(Pkl) ∩ Pnj 6= ∅. (2)
Let g ∈ Xn be such that
g(Pni) = Pni, (3)
g(Pnj) 6= Pnj . (4)
Since gγ ∈ Xk, we get gγ(Pkl) = Pkm, where 1 ≤ m ≤ fΩ(k). Taking
into account (1) and (3) we get gγ(Pkl) = Pkl and by gγ ∈ Xk we have
gγ(x) = x for all x ∈ Pkl.
Also taking into account (2) and (4) we get that there exists x0 ∈
Pkl ∩ γ(Pnj) such that gγ(x0) 6= x0. This is a contradiction. Hence,
γ−1(Pkl) ⊂ Pni for some i such that 1 ≤ i ≤ fΩ(n).
Let
γ(Pni) = Pkli1 t . . . t Pkli,r(i)
for some r(i), where 1 ≤ li1, . . . , li,r(i) ≤ fΩ(k).
Since Xn is transitive, we can choose gi ∈ Xn, 1 ≤ i ≤ fΩ(n) such
that gi(Pni) = Pn1. Since gγ
i ∈ Xk and the sets γ(Pni) and γ(Pnj) do not
intersect, there exists a mapping
t : {1, . . . , r(i)} × {1, . . . , fΩ(n)} → {1, . . . , r(1)}
such that for every m and i, 1 ≤ m ≤ r(i), 1 ≤ i ≤ fΩ(n), the following
equality holds
gi
(
P γ−1
klim
)
= P γ−1
kl1,t(m,i)
. (5)
Note that t(m, i) does not depend on the choice of gi. Obviously, t(m, i)
is a bijection for every fixed i. Thus r(i) does not depend on i for 1 ≤
i ≤ fΩ(n). It is easy to see that r = r(i) = fΩ(n)/fΩ(n − 1) for all
1 ≤ i ≤ fΩ(n).
Let us define a homeomorphism π ∈ Hk as follows: the homeomor-
phism π maps the vertex vklim corresponding to the ball Pklim to the
vertex g−1
i (vk,t(m,i)) corresponding to the ball g−1
i (Pk,t(m,i)), for every i,
1 ≤ i ≤ fΩ(n), and for every m, 1 ≤ m ≤ r. Here t(m, i) is as in (5).
Since gi ∈ Xn, g−1
i (Pn1) = Pni and the vertex vk,t(m,i) lies under the
vertex vn1, the homeomorphism π is well-defined and does not depend
on the choice of gi. We remind that gi is an element of Xn such that
gi(Pni) = Pn1.
Let g ∈ Xn such that g(Pni) = Pnj . According to (5), we get
g
(
P γ−1
kl
i,t−1(m,i)
)
= P γ−1
kl
j,t−1(m,j)
. (6)
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.92 A note to our paper “Automorphisms of homogeneous...”
Obviously g′ = πγgγ−1π−1 belongs to Hk. Let Pks ⊂ Pni. Using (6), we
get
g′(Pks) = πγgγ−1
(
Pkl
i,t−1(s+r−ir,i)
)
=
= π
(
Pkl
j,t−1(s+r−ir,j)
)
= Pk,s+(j−i)r = g(Pks).
Therefore πγ centralizes Xn. Hence, gγ = gπ−1
for all g ∈ Xn.
So we have proved that α|Xn
is induced by an inner automorphism
of Hk.
The main result of this note is then stated as follows.
Theorem 2. The automorphism group of the subgroup AHΩ coincides
with the automorphism group of the group HΩ.
Proof. Let α be an automorphism of AHΩ. The relation α|AHn
∈
InnHk follows from Proposition 1. The last inclusion implies that
α ∈ NHomeo(∂TΩ)(HΩ). Hence Aut AHΩ ≤ Aut HΩ. On the other hand,
AHΩ is the commutant of HΩ. Thus Aut HΩ is a subgroup of Aut AHΩ.
So Aut AHΩ = AutHΩ.
We also correct a misprint in [LS]. On p. 38, line 7 of [LS], the phrase
“we have Pni ∩Pmj 6= ∅ if and only if Pni = Pmj that is n = m and i = j”
should be replaced by “we have Pni ∩ Pnj 6= ∅ if and only if Pni = Pnj ,
that is i = j”.
References
[LS] Ya.V.Lavrenyuk, V.I.Sushchansky Automorphisms of Homogeneous Symmetric
Groups and Hierarchomorphisms of Rooted Trees, Algebra and Discrete Mathe-
matics, N. 4 (2003), 33–49.
[Rub] Rubin M. On the reconstruction of topological spaces from their groups of home-
omorphisms. Transactions of the AMS, Vol. 312, No 2. April 1989, 487-538.
Contact information
Yaroslav Lavrenyuk Department of Mechanics and Mathematics,
Kyiv Taras Shevchenko University, 64, Vo-
lodymyrska st., 01033, Kyiv, Ukraine
E-Mail: ylavrenyuk@univ.kiev.ua
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.Y. V. Lavrenyuk, V. I. Sushchansky 93
Vitaly Sushchansky Institute of Mathematics, Silesian Univer-
sity of Technology, 23, ul. Kaszubska, 44-
100, Gliwice, Poland
E-Mail: Wital.Suszczanski@polsl.pl
Received by the editors: 30.05.2005
and final form in 05.07.2005.
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