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On commuting probabilities in finite groups and rings

Year 2022, Volume: 9 Issue: 1, 9 - 15, 15.01.2022

Martin Juras Mihail Ursul

https://doi.org/10.13069/jacodesmath.1056492

Abstract

We show that the set of all commuting probabilities in finite rings is a subset ofthe set of all commuting probabilities in finite nilpotent groups of class $\leq 2$. These two sets are equal when restricted to groups and rings with odd number of elements.

Keywords

Finite group Finite ring Commuting probability Annihilating probability Nilpotent group Nilpotent ring

References

  • [1] S. M. Buckley, D. MacHale, Y. Zelenyuk, Finite rings with large anticommuting probability, Appl. Math. Inf. Sci. 8(1) (2014) 13–25.
  • [2] S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, Contemp. Math. 634 (2015) 31–52.
  • [3] S. M. Buckley, D. MacHale, Commuting probability for subrings and quotient rings, J. Algebra Comb. Discrete Appl. 4(2) (2017) 189–196.
  • [4] S. M. Buckley, D. MacHale, Contrasting the commuting probabilities of groups and rings, preprint.
  • [5] S. M. Buckley, D. MacHale, Á. N. Shé, Finite rings with many commuting pairs of elements, preprint.
  • [6] A. K. Das, R. K. Nath, A characterisation of certain finite groups of odd order, Math. Proc. R. Ir. Acad. 111(2) (2011) 67–76.
  • [7] J. Dixon, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24(1) (2002) 1–15.
  • [8] P. Erdös, P. Turán, On some problems of a statistical group-theory. I, Z. Wahrschein. Verw. Gebiete. 4 (1965) 175–186.
  • [9] P. Erdös, P. Turán, On some problems of a statistical group-theory. II, Acta Math. Acad. Sci. Hungar. 18 (1967) 151–163.
  • [10] P. Erdös, P. Turán, On some problems of a statistical group-theory. III, Acta Math. Acad. Sci. Hungar. 18 (1967) 309–320.
  • [11] P. Erdös, P. Turán, On some problems in statistical group-theory. IV, Acta Math. Acad. Sci. Hungar. 19 (1968) 413–453.
  • [12] W. Feit, N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. J. 27(1) (1960) 91–94.
  • [13] P. X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970) 175–179.
  • [14] B. Givens, The probability that two semigroup elements commute can be almost anything. College Mathematics Journal 39(5) (2008) 399–400.
  • [15] R. M. Guralnick, G. R. Robinson, On commuting probability in finite groups, J. Algebra 300(2) (2006) 509–528.
  • [16] W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80(9) (1973) 1031–1034.
  • [17] P. Hall, The classification of prime-power groups, J. Reine. Agnew. Math. 182 (1940) 130–141.
  • [18] P. Hegarty, Limit points in the range of the commuting probability function on finite groups, J. Group Theory 16 (2013) 235–247.
  • [19] K. S. Joseph, Commutativity in non-abelian groups, Ph.D. Thesis, University of California, Los Angeles (1969).
  • [20] K. S. Joseph, Several conjectures on commutativity in algebraic structures, Amer. Math. Monthly 84(7) (1977) 550–551.
  • [21] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177(3) (1995) 847–869.
  • [22] D. MacHale, How commutative can a non-commutative group be?, Math. Gaz. 58 (1974) 199–202.
  • [23] D. MacHale, Commutativity in finite rings, Amer. Math. Monthly 84(1) (1976) 30-32.
  • [24] D. MacHale, Probability in fnite semigroups, Irish Math. Soc. Bull. 25 (1990) 64–68.
  • [25] A. I. Mal’cev, On a correspondence between rings and groups, in Fifteen papers on algebra, AMS translation American Mathematical Soc. (1965) 221–232.
  • [26] V. Ponomarenko, N. Selinski, Two semigroup elements can commute with any positive rational probability, College Math. J. 43(4) (2012) 334–336.
  • [27] D. J. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82(1) (1979) 237–247.
  • [28] Á. N. Shé, Commutativity and generalizations in finite groups, Ph.D. Thesis, University College Cork (2000).
  • [29] M. Soule, A single family of semigroups with every positive rational commuting probability, College Math. J. 45(2) (2014) 136–139.

Year 2022, Volume: 9 Issue: 1, 9 - 15, 15.01.2022

Martin Juras Mihail Ursul

https://doi.org/10.13069/jacodesmath.1056492

Abstract

References

  • [1] S. M. Buckley, D. MacHale, Y. Zelenyuk, Finite rings with large anticommuting probability, Appl. Math. Inf. Sci. 8(1) (2014) 13–25.
  • [2] S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, Contemp. Math. 634 (2015) 31–52.
  • [3] S. M. Buckley, D. MacHale, Commuting probability for subrings and quotient rings, J. Algebra Comb. Discrete Appl. 4(2) (2017) 189–196.
  • [4] S. M. Buckley, D. MacHale, Contrasting the commuting probabilities of groups and rings, preprint.
  • [5] S. M. Buckley, D. MacHale, Á. N. Shé, Finite rings with many commuting pairs of elements, preprint.
  • [6] A. K. Das, R. K. Nath, A characterisation of certain finite groups of odd order, Math. Proc. R. Ir. Acad. 111(2) (2011) 67–76.
  • [7] J. Dixon, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24(1) (2002) 1–15.
  • [8] P. Erdös, P. Turán, On some problems of a statistical group-theory. I, Z. Wahrschein. Verw. Gebiete. 4 (1965) 175–186.
  • [9] P. Erdös, P. Turán, On some problems of a statistical group-theory. II, Acta Math. Acad. Sci. Hungar. 18 (1967) 151–163.
  • [10] P. Erdös, P. Turán, On some problems of a statistical group-theory. III, Acta Math. Acad. Sci. Hungar. 18 (1967) 309–320.
  • [11] P. Erdös, P. Turán, On some problems in statistical group-theory. IV, Acta Math. Acad. Sci. Hungar. 19 (1968) 413–453.
  • [12] W. Feit, N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. J. 27(1) (1960) 91–94.
  • [13] P. X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970) 175–179.
  • [14] B. Givens, The probability that two semigroup elements commute can be almost anything. College Mathematics Journal 39(5) (2008) 399–400.
  • [15] R. M. Guralnick, G. R. Robinson, On commuting probability in finite groups, J. Algebra 300(2) (2006) 509–528.
  • [16] W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80(9) (1973) 1031–1034.
  • [17] P. Hall, The classification of prime-power groups, J. Reine. Agnew. Math. 182 (1940) 130–141.
  • [18] P. Hegarty, Limit points in the range of the commuting probability function on finite groups, J. Group Theory 16 (2013) 235–247.
  • [19] K. S. Joseph, Commutativity in non-abelian groups, Ph.D. Thesis, University of California, Los Angeles (1969).
  • [20] K. S. Joseph, Several conjectures on commutativity in algebraic structures, Amer. Math. Monthly 84(7) (1977) 550–551.
  • [21] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177(3) (1995) 847–869.
  • [22] D. MacHale, How commutative can a non-commutative group be?, Math. Gaz. 58 (1974) 199–202.
  • [23] D. MacHale, Commutativity in finite rings, Amer. Math. Monthly 84(1) (1976) 30-32.
  • [24] D. MacHale, Probability in fnite semigroups, Irish Math. Soc. Bull. 25 (1990) 64–68.
  • [25] A. I. Mal’cev, On a correspondence between rings and groups, in Fifteen papers on algebra, AMS translation American Mathematical Soc. (1965) 221–232.
  • [26] V. Ponomarenko, N. Selinski, Two semigroup elements can commute with any positive rational probability, College Math. J. 43(4) (2012) 334–336.
  • [27] D. J. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82(1) (1979) 237–247.
  • [28] Á. N. Shé, Commutativity and generalizations in finite groups, Ph.D. Thesis, University College Cork (2000).
  • [29] M. Soule, A single family of semigroups with every positive rational commuting probability, College Math. J. 45(2) (2014) 136–139.
There are 29 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Martin Juras This is me 0000-0003-4752-7734

Mihail Ursul This is me 0000-0003-4744-0890

Early Pub Date January 31, 2022
Publication Date January 15, 2022
Published in Issue Year 2022 Volume: 9 Issue: 1

Cite

APA Juras, M., & Ursul, M. (n.d.). On commuting probabilities in finite groups and rings. Journal of Algebra Combinatorics Discrete Structures and Applications, 9(1), 9-15. https://doi.org/10.13069/jacodesmath.1056492
AMA Juras M, Ursul M. On commuting probabilities in finite groups and rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 9(1):9-15. doi:10.13069/jacodesmath.1056492
Chicago Juras, Martin, and Mihail Ursul. “On Commuting Probabilities in Finite Groups and Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 9, no. 1 n.d.: 9-15. https://doi.org/10.13069/jacodesmath.1056492.
EndNote Juras M, Ursul M On commuting probabilities in finite groups and rings. Journal of Algebra Combinatorics Discrete Structures and Applications 9 1 9–15.
IEEE M. Juras and M. Ursul, “On commuting probabilities in finite groups and rings”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 1, pp. 9–15, doi: 10.13069/jacodesmath.1056492.
ISNAD Juras, Martin - Ursul, Mihail. “On Commuting Probabilities in Finite Groups and Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 9/1 (n.d.), 9-15. https://doi.org/10.13069/jacodesmath.1056492.
JAMA Juras M, Ursul M. On commuting probabilities in finite groups and rings. Journal of Algebra Combinatorics Discrete Structures and Applications.;9:9–15.
MLA Juras, Martin and Mihail Ursul. “On Commuting Probabilities in Finite Groups and Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 1, pp. 9-15, doi:10.13069/jacodesmath.1056492.
Vancouver Juras M, Ursul M. On commuting probabilities in finite groups and rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 9(1):9-15.