Groups Generated by (near) Mutually Engel Periodic Pairs

Piotr Słanina; Witold Tomaszewski

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 2, page 485-497
  • ISSN: 0392-4033

Abstract

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We use notations: [ x , y ] = [ x , 1 y ] and [ x , k + 1 y ] e [ [ x , k y ] , y ] . We consider groups generated by x , y satisfying relations x = [ x , n y ] , y = [ y , n x ] or [ x , y ] = [ x , n y ] , [ y , x ] = [ y , n x ] . We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if p is a prime then the group S l 2 ( p ) is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation x , y [ x , y ] = [ x , 2 y ] , [ y , x ] = [ y , 2 x ] , x n , y m are finite.

How to cite

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Słanina, Piotr, and Tomaszewski, Witold. "Groups Generated by (near) Mutually Engel Periodic Pairs." Bollettino dell'Unione Matematica Italiana 10-B.2 (2007): 485-497. <http://eudml.org/doc/290362>.

@article{Słanina2007,
abstract = {We use notations: $[x, y]=[x_\{,1\} y]$ and $[x_\{,k+1\} y]$ e $[[x_\{,k\} y], y]$. We consider groups generated by $x$, $y$ satisfying relations $x = [x_\{,n\} y], y = [y_\{,n\} x]$ or $[x, y]=[x_\{,n\} y]$, $[y, x]=[y_\{,n\} x]$. We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if $p$ is a prime then the group $Sl_2(p)$ is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation $\langle x,y \mid [x,y] = [x_\{,2\} y], [y,x]=[y_\{,2\} x], x^n, y^m \rangle$ are finite.},
author = {Słanina, Piotr, Tomaszewski, Witold},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {485-497},
publisher = {Unione Matematica Italiana},
title = {Groups Generated by (near) Mutually Engel Periodic Pairs},
url = {http://eudml.org/doc/290362},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Słanina, Piotr
AU - Tomaszewski, Witold
TI - Groups Generated by (near) Mutually Engel Periodic Pairs
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/6//
PB - Unione Matematica Italiana
VL - 10-B
IS - 2
SP - 485
EP - 497
AB - We use notations: $[x, y]=[x_{,1} y]$ and $[x_{,k+1} y]$ e $[[x_{,k} y], y]$. We consider groups generated by $x$, $y$ satisfying relations $x = [x_{,n} y], y = [y_{,n} x]$ or $[x, y]=[x_{,n} y]$, $[y, x]=[y_{,n} x]$. We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if $p$ is a prime then the group $Sl_2(p)$ is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation $\langle x,y \mid [x,y] = [x_{,2} y], [y,x]=[y_{,2} x], x^n, y^m \rangle$ are finite.
LA - eng
UR - http://eudml.org/doc/290362
ER -

References

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  1. COXETER, H.S.M., MOSER, W.O.J., Generators and Relations for Discrete Groups, Berlin-Heildeberg-New York1980. Zbl0422.20001MR562913
  2. HEINEKEN, H., Groups generated by two mutually Engel Periodic elements, Bolletino U.M.I., (8) 3-B (2000), 461-470. Zbl0982.20016MR1769996
  3. KARGAPOLOV, M.I., MERZLJAKOV, JU. I., Fundamentals of the Theory of Groups, Springer-Verlag, New York, 1979. Zbl0549.20001MR551207
  4. MAGNUS, W., KARRASS, A., SOLITAR, D., Combinatorial Group Theory, Dover Publications, Inc. New York1976. MR422434
  5. ROTMAN, J.J., An Introduction to the Theory of Groups, Springer-Verlag, Inc. New York1995. Zbl0810.20001MR1307623DOI10.1007/978-1-4612-4176-8
  6. ZASSENHAUS, H.J., The Theory of Groups, Dover Publications, Inc., Mineola, New York1999. Zbl0943.20002MR1644892

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