On commuting probabilities in finite groups and rings
DOI:
https://doi.org/10.13069/jacodesmath.1056492Keywords:
Finite group, Finite ring, Commuting probability, Annihilating probability, Nilpotent group, Nilpotent ringAbstract
We show that the set of all commuting probabilities in finite rings is a subset of the 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.
Received: 25 May 2020 | Accepted: 30 September 2021Downloads
References
S. M. Buckley, D. MacHale, and Y. Zelenyuk, Finite rings with large anticommuting probability, Appl. Math. Inf. Sci. 8(1) (2014) 13–25.
S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, Contemp. Math. 634 (2015) 31–52.
S. M. Buckley and D. MacHale, Commuting probability for subrings and quotient rings, J. Algebra Comb. Discrete Appl. 4(2) (2017) 189–196.
S. M. Buckley and D. MacHale, Contrasting the commuting probabilities of groups and rings, preprint.
S. M. Buckley, D. MacHale, and Á. N. Shé, Finite rings with many commuting pairs of elements, preprint.
A. K. Das and R. K. Nath, A characterisation of certain finite groups of odd order, Math. Proc. R. Ir. Acad. 111(2) (2011) 67–76.
J. D. Dixon, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24(1) (2002) 1–15.
P. Erdös and P. Turán, On some problems of a statistical group-theory. I, Z. Wahrschein. Verw. Gebiete 4 (1965) 175–186.
P. Erdös and P. Turán, On some problems of a statistical group-theory. II, Acta Math. Acad. Sci. Hungar. 18 (1967) 151–163.
P. Erdös and P. Turán, On some problems of a statistical group-theory. III, Acta Math. Acad. Sci. Hungar. 18 (1967) 309–320.
P. Erdös and P. Turán, On some problems in statistical group-theory. IV, Acta Math. Acad. Sci. Hungar. 19 (1968) 413–453.
W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. J. 27(1) (1960) 91–94.
P. X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970) 175–179.
B. Givens, The probability that two semigroup elements commute can be almost anything, College Math. J. 39(5) (2008) 399–400.
R. M. Guralnick and G. R. Robinson, On the commuting probability in finite groups, J. Algebra 300(2) (2006) 509–528.
W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80(9) (1973) 1031–1034.
P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940) 130–141.
P. Hegarty, Limit points in the range of the commuting probability function on finite groups, J. Group Theory 16(2) (2013) 235–247.
K. S. Joseph, Commutativity in non-abelian groups, Ph.D. Thesis, University of California, Los Angeles (1969).
K. S. Joseph, Several conjectures on commutativity in algebraic structures, Amer. Math. Monthly 84(7) (1977) 550–551.
P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177(3) (1995) 847–869.
D. MacHale, How commutative can a non-commutative group be?, Math. Gaz. 58(405) (1974) 199–202.
D. MacHale, Commutativity in finite rings, Amer. Math. Monthly 83(1) (1976) 30–32.
D. MacHale, Probability in finite semigroups, Irish Math. Soc. Bull. 25 (1990) 64–68.
A. I. Mal'cev, On a correspondence between rings and groups, Amer. Math. Soc. Transl. (2) 45 (1965) 221–232.
V. Ponomarenko and N. Selinski, Two semigroup elements can commute with any positive rational probability, College Math. J. 43(4) (2012) 334–336.
D. J. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82(1) (1979) 237–247.
Á. N. Shé, Commutativity and generalizations in finite groups, Ph.D. Thesis, University College Cork (2000).
M. Soule, A single family of semigroups with every positive rational commuting probability, College Math. J. 45(2) (2014) 136–139.
