SOME CONSEQUENCES OF THE EQUATION [xn, y] = 1 ON THE STRUCTURE OF A COMPACT GROUP

Title & Authors
SOME CONSEQUENCES OF THE EQUATION [xn, y] = 1 ON THE STRUCTURE OF A COMPACT GROUP
Erfanian, Ahmad; Rezaei, Rashid; Tolue, Behnaz;

Abstract
Given an integer $\small{n{\geq}1}$ and a compact group G, we find some restrictions for the probability that two randomly picked elements $\small{x^n}$ and $\small{y}$ of G commute. In the case $\small{n=1}$ this notion was investigated by W. H. Gustafson in 1973 and its influence on the structure of the group has been studied in the researches of several authors in last years.
Keywords
n-th power central elements;commutativity degree;compact groups;
Language
English
Cited by
References
1.
M. Abert, On the probability of satisfying a word in a group, Cornell University Library, preprint, 2005, available online at http://arxiv.org/abs/math/0504312.

2.
A.M. Alghamdi and F. G. Russo, A generalization of the probability that the commutator of two group elements is equal to a given element, Bull. Iranian Math. Soc., in press.

3.
N. M. Ali and N. Sarmin, On some problems in group theory of probabilistic nature, Technical Report, Universiti Teknologi Malaysia, Johor Bahru, Malaysia, 2009.

4.
R. Brandl and L.-C. Kappe, On n-Bell groups, Comm. Algebra 17 (1989), no. 4, 787- 807.

5.
K. Chiti, M. R. R. Moghaddam, and A. R. Salemkar, n-isoclinism classes and n- nilpotency degree of finite groups, Algebra Colloq. 12 (2005), no. 2, 255-261.

6.
P. Erd˝os and P. Turan, On some problems of a statistical group-theory. IV, Acta Math. Acad. Sci. Hungar 19 (1968), 413-435.

7.
A. Erfanian and R. Rezaei, On the commutativity degree of compact groups, Arch. Math. (Basel) 93 (2009), no. 4, 201-212.

8.
A. Erfanian and F. G. Russo, Probability of mutually commuting n-tuples in some classes of compact groups, Bull. Iranian Math. Soc. 34 (2008), no. 2, 27-37.

9.
A. Erfanian, B. Tolue, and N. Sarmin, Some consideration on the n-th commutativity degrees of finite groups, Ars. Comb. (2010), in press.

10.
P. X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970), 175-179.

11.
W. H. Gustafson, What is the probability that two groups elements commute?, Amer. Math. Monthly 80 (1973), 1031-1034.

12.
P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130-141.

13.
K. H. Hofmann and S. A. Morris, The Structure of Compact Groups, de Gruyter, Berlin, 2006.

14.
K. H. Hofmann, S. Morris, and M. Stroppel, Locally compact groups, residual Lie groups, and varieties generated by Lie groups, Topology Appl. 71 (1996), no. 1, 63-91.

15.
K. H. Hofmann and F. G. Russo, The probability that x and y commute in a compact group, Cornelly University Library, 2010, available online at http://arxiv.org/ abs/1001.4856.

16.
P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177 (1995), no. 3, 847-869.

17.
M. Levy, On the probability of satisfying a word in nilpotent groups of class 2, Cornell University Library, preprint, 2011, available online at http://arxiv.org/abs/1101.4286.

18.
Y. Medvedev, On compact Engel groups, Israel J. Math. 135 (2003), 147-156.

19.
H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford, Clarendon Press, 1968.

20.
R. Rezaei and F. G. Russo, Bounds for the relative n-th nilpotency degree in compact groups, Asian-Eur. J. Math. 4 (2011), no. 3, 495-506.

21.
R. Rezaei and F. G. Russo, n-th relative nilpotency degree and relative n-isoclinism classes, Carpathian J. Math. 27 (2011), no. 1, 123-130.

22.
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer-Verlag, Berlin, 1972.

23.
D. Segal, Words: Notes on verbal width in groups, LMS Lecture Notes Serie 361, Cambridge University Press, Cambridge, 2009.

24.
J. Wiegold, Multiplicators and groups with finite central factor-groups, Math. Z. 89 (1965), 345-347.