# 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

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topSłanina, 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

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

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