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Chebyshevskii Sb., 2016, Volume 17, Issue 2, Pages 113–127 (Mi cheb482)  

This article is cited in 2 scientific papers (total in 2 papers)

On normalizers in some Coxeter groups

I. V. Dobrynina

Tula State Pedagogical University

Abstract: Let $G$ be a finitely generated Coxeter group with presentation
$$G=< a_1,\ldots, a_n;(a_ia_j)^{m_{ij}}=1,   i,j =\overline{1,n} >,$$
where $m_{ij}$ — are the elements of the symmetric Coxeter matrix: $\forall i,j \in\overline{1,n},  m_{ii}=1, m_{ij} \geq$ $ \geq2,   i\ne j$.
If $m_{ij}\geq3$ $(m_{ij}>3)$, $i\ne j$, then $G$ is a Coxeter group of large (extra-large) type. These groups introduced by K. Appel and P. Schupp.
If the group $G$ corresponds to a finite tree-graph $\Gamma$ such that if the vertices of some edge $e$ of the graph $\Gamma$ correspond to generators $a_i, a_j$, then the edge $e$ corresponds to the ratio of the species $(a_ia_j)^{m_{ij}}=1$, then $G$ is a Coxeter group with a tree-structure.
Coxeter groups with a tree-structure introduced by V. N. Bezverkhnii, algorithmic problems in them was considered by V. N. Bezverkhnii and O. V. Inchenko.
The group $G$ can be represented as tree product 2-generated of Coxeter groups, amalgamated by cyclic subgroups. Thus from the graph $\Gamma$ of $G$ will move to the graph $\overline{\Gamma}$ in the following way: the vertices of the graph $\overline{\Gamma}$ we will put in line Coxeter group on two generators
$$G_{ij} = <a_i, a_j; a_i^2=a_j^2=1,(a_ia_j)^{m_{ij}}=1>$$
and
$$G_{jk} = <a_j, a_k; a_j^2=a_k^2=1,(a_ja_k)^{m_{jk}}=1>,$$
to every edge $\overline{e}$ joining the vertices corresponding to $G_{ij}$ and $G_{jk}$ is a cyclic subgroup
$$<a_j;a_j^2=1>.$$

In this paper we prove the following theorem: normalizer of finitely generated subgroup of Coxeter group with tree-structure
$$\overline{G}=G_{ij}\ast_{<a_j; a_j^2>}G_{jk},$$

$$G_{ij} = <a_i, a_j; a_i^2=a_j^2=1,(a_ia_j)^{m_{ij}}=1>,$$

$$G_{jk} = <a_j, a_k; a_j^2=a_k^2=1,(a_ja_k)^{m_{jk}}=1>$$
finitely generated and exist algorithm for generating.
Bibliography: 18 titles.

Keywords: Coxeter group, tree-structure, normalizer, amalgamated product.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-41-03222_р_центр_а


Full text: PDF file (662 kB)
References: PDF file   HTML file
UDC: 519.4
Received: 16.04.2016
Accepted:10.06.2016

Citation: I. V. Dobrynina, “On normalizers in some Coxeter groups”, Chebyshevskii Sb., 17:2 (2016), 113–127

Citation in format AMSBIB
\Bibitem{Dob16}
\by I.~V.~Dobrynina
\paper On normalizers in some Coxeter groups
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 2
\pages 113--127
\mathnet{http://mi.mathnet.ru/cheb482}
\elib{http://elibrary.ru/item.asp?id=26254427}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. N. Bezverkhnii, N. B. Bezverkhnyaya, I. V. Dobrynina, O. V. Inchenko, A. E. Ustyan, “Ob algoritmicheskikh problemakh v gruppakh Kokstera”, Chebyshevskii sb., 17:4 (2016), 23–50  mathnet  crossref  elib
    2. I. V. Dobrynina, “O normalizatorakh podgrupp v gruppakh Kokstera s drevesnoi strukturoi”, Sib. elektron. matem. izv., 14 (2017), 1338–1348  mathnet  crossref
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