On commuting probabilities in finite groups and rings

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. [1] S. M. Buckley, D. MacHale, Y. Zelenyuk, Finite rings with large anticommuting probability, Appl. Math. Inf. Sci. 8(1) (2014) 13–25.
2. [2] S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, Contemp. Math. 634 (2015) 31–52.
3. [3] S. M. Buckley, D. MacHale, Commuting probability for subrings and quotient rings, J. Algebra Comb. Discrete Appl. 4(2) (2017) 189–196.
4. [4] S. M. Buckley, D. MacHale, Contrasting the commuting probabilities of groups and rings, preprint.
5. [5] S. M. Buckley, D. MacHale, Á. N. Shé, Finite rings with many commuting pairs of elements, preprint.
6. [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. [7] J. Dixon, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24(1) (2002) 1–15.
8. [8] P. Erdös, P. Turán, On some problems of a statistical group-theory. I, Z. Wahrschein. Verw. Gebiete. 4 (1965) 175–186.

9. [9] P. Erdös, P. Turán, On some problems of a statistical group-theory. II, Acta Math. Acad. Sci. Hungar. 18 (1967) 151–163.
10. [10] P. Erdös, P. Turán, On some problems of a statistical group-theory. III, Acta Math. Acad. Sci. Hungar. 18 (1967) 309–320.
11. [11] P. Erdös, P. Turán, On some problems in statistical group-theory. IV, Acta Math. Acad. Sci. Hungar. 19 (1968) 413–453.
12. [12] W. Feit, N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. J. 27(1) (1960) 91–94.
13. [13] P. X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970) 175–179.
14. [14] B. Givens, The probability that two semigroup elements commute can be almost anything. College Mathematics Journal 39(5) (2008) 399–400.
15. [15] R. M. Guralnick, G. R. Robinson, On commuting probability in finite groups, J. Algebra 300(2) (2006) 509–528.
16. [16] W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80(9) (1973) 1031–1034.
17. [17] P. Hall, The classification of prime-power groups, J. Reine. Agnew. Math. 182 (1940) 130–141.
18. [18] P. Hegarty, Limit points in the range of the commuting probability function on finite groups, J. Group Theory 16 (2013) 235–247.
19. [19] K. S. Joseph, Commutativity in non-abelian groups, Ph.D. Thesis, University of California, Los Angeles (1969).
20. [20] K. S. Joseph, Several conjectures on commutativity in algebraic structures, Amer. Math. Monthly 84(7) (1977) 550–551.
21. [21] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177(3) (1995) 847–869.
22. [22] D. MacHale, How commutative can a non-commutative group be?, Math. Gaz. 58 (1974) 199–202.
23. [23] D. MacHale, Commutativity in finite rings, Amer. Math. Monthly 84(1) (1976) 30-32.
24. [24] D. MacHale, Probability in fnite semigroups, Irish Math. Soc. Bull. 25 (1990) 64–68.
25. [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. [26] V. Ponomarenko, N. Selinski, Two semigroup elements can commute with any positive rational probability, College Math. J. 43(4) (2012) 334–336.
27. [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. [28] Á. N. Shé, Commutativity and generalizations in finite groups, Ph.D. Thesis, University College Cork (2000).
29. [29] M. Soule, A single family of semigroups with every positive rational commuting probability, College Math. J. 45(2) (2014) 136–139.