ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1056492 J. Algebra Comb. Discrete Appl. 9(1) • 9–15 Received: 25 May 2020 Accepted: 30 September 2021 Journal of Algebra Combinatorics Discrete Structures and Applications On commuting probabilities in finite groups and rings Research Article Martin Juráš, Mihail Ursul Abstract: 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 ≤ 2. These two sets are equal when restricted to groups and rings with odd number of elements. 2010 MSC: 16U80, 05C25, 20P05, 16N40, 20D15 Keywords: Finite group, Finite ring, Commuting probability, Annihilating probability, Nilpotent group, Nilpotent ring 1. Introduction and preliminaries In 1940, Philip Hall [17] introduced the notion of the commuting probability in groups. Feit and Fine [12], derived a combinatorial formula and a generating function for commuting probability in matrix rings over finite fields. In the second half of 1960’s, the series of papers [8], [9], [10], [11] by Erdös and Turán, gave birth to the statistical group theory. In the fourth paper, among other results, the authors derived a lower bound for commuting probability in a finite group of order n, and showed that the commuting probability in the symmetric group Sn is asymptotically equal to 1n· A number of research and expository papers on commuting probability in groups appeared during late sixties and the seventies: Joseph [19], [20], Galagher [13], Gustafson [16], Machale [22], and Rusin [27], to name a few1. Rusin [27], characterized all finite groups with commuting probability > 11 32 · There has also been interest in the study of commuting probability of other algebraic structures, [20]. MacHale [23], investigated the notion of commuting probability in rings. In 1995 Lescot [21], re- derived classification of groups with commuting probability > 1 2 , using the notion of isoclinism in groups introduced by Hall [17]. Recently, the commuting probability in semigroups has been studied in [14], [24], [26] and [29]. Martin Juráš (Corresponding Author); SCAD, Savannah, GA 31401, USA (email: mjuras@scad.edu, martin- juras@gmail.com). Mihail Ursul; Department of Mathematics, PNG University of Technology, Lae, PNG (email: mihail.ursul@ gmail.com). 1 Dixon, provides an extensive list of publications on statistical group theory in the references of his paper [7], up to the year 2002. 9 https://orcid.org/0000-0003-4752-7734 https://orcid.org/0000-0003-4744-0890 M. Juráš, M. Ursul / J. Algebra Comb. Discrete Appl. 9(1) (2022) 9–15 Since the dawn of the twenty-first century we have seen an escalation of interest in the study of the commuting probability in groups, and commuting and other types of probabilities in rings, such as anticommuting and annihilating probability. Publications [28], [7], [15], [6] and [18] deal with commuting probability in groups. In papers [3] and [1], Buckley et. al. classified all rings with commuting probability ≥ 11 32 and anticommuting probability ≥ 15 32 , respectively. Throughout this paper, |A| denotes cardinality of the set A. Z(G) denotes the center of a group G. For a, b ∈ G, [a, b] = a−1b−1ab denotes the commutator of a and b, and [G, G] denotes the derived subgroup of G generated by all commutators in G. Recall that G is nilpotent of class n, if its lower central series (of normal subgroups) terminate in the trivial subgroup after n steps, i.e. G = G0 . G1 . · · ·. Gn = {eG}, where Gi = [Gi−1, G] for i = 1, 2, . . . , n, and Gn−1 6= {eG}. Commuting probability2 in a group G is defined to be the number Prc(G) = |{(a, b) ∈ G×G : ab = ba}| |G|2 . For a class G of finite groups, the set Sc(G) = {Prc(G) : G ∈G} is called the commuting spectrum of G. Rings are not assumed to be associative or unitary. By R(+) we denote the additive group of R. Recall that a ring R is called antisymmetric if for all a, b ∈ R, ab = −ba. R is called strongly antisymmetric if the dinipotent condition, a2 = 0, is satisfied for all a ∈ R. Strong antisymmetry implies antisymmetry. A ring R is said to be of nilpotent class ≤ n if the product of any n elements with any correct distribution of brackets is zero. For a prime p, R is called a p-ring if |R| = pn for some positive integer n. The symbol [·, ·] denotes the commutator in both a group G and a ring R (for rings, [a, b] = ab−ba). Whenever needed, we will write [·, ·]G and [·, ·]R to distinguish between the two cases. Buckley [2], introduced the following generalization of the notion of commuting probability in rings. Let f(X, Y ) = aXY + bY X be a formal "non-commutative polynomial" with integer coefficients. For any ring R define a function fR : R ×R → R, (x, y) = axy + byx. Let Prf (R) = |{(x, y) ∈ R ×R : fR(x, y) = 0}| |R|2 · For a class R of finite rings, the set Sf (R) = {Prf (R) : R ∈R} is called the f-spectrum of R. Here, we are going to be mostly concerned with the commuting spectrum, Sc(R) and the annihilating spectrum, Sann(R), with the associated formal "non-commutative polynomials" f(X, Y ) = XY − Y X and f(X, Y ) = XY , respectively. The commuting probability and the annihilating probability in a ring R are denoted by Prc(R) and Prann(R), respectively. We will use the following classes of groups and rings: G the class of finite groups; Gnil the class of finite nilpotent groups; G (2) nil the class of finite nilpotent groups of class ≤ 2; R the class of finite rings; R (2) nil the class of finite nilpotent rings of class ≤ 3; 2 Some publications use the term commuting degree in place of the commuting probability. 10 M. Juráš, M. Ursul / J. Algebra Comb. Discrete Appl. 9(1) (2022) 9–15 Rsa the class of finite strongly antisymmetric rings; Rp the class of p-rings; for the class C of finite sets, denote ODD(C) = {A ∈C : |A| is odd}. Recall the following well know construction. For a given ring R, we construct the ring N(R) in the following way: the additive group of N(R) is (R × R, +) with multiplication (a, x)(b, y) = (0, ab). The following Lemma is immediate. Lemma 1.1. Let R be a ring. Then N(R) is a nilpotent ring of class at most 3. Furthermore, if f(X, Y ) = aXY + bY X is a formal non-commutative polynomial with integer coefficients and R is finite, then Prf (R) = Prf (N(R)). In particular, the Lemma implies Sf (R) = Sf (R (2) nil). (1) Ever since it was discovered that there are no finite groups with commuting probability in the open interval 5 8 ), 1), there has been an interest to understand the structure of the commuting spectrum of groups, and later, the structure of the commuting spectrum of rings and semigroups. The commuting spectrum for semigroups turned out to be the simplest to understand. Givens [14] showed that the commuting spectrum for semigroups is dense in the interval [0, 1]. Later Ponomarenko and Selinski [26] proved that for any rational number in r ∈ (0, 1], there is a finite semigroup S such that the commuting probability in S is equal to r. Soule [29] found a single family of semigroups that has this property. These semigroups are defined as follows. Let X = {x1, x2, . . . , xm}∈ {1, 2, . . . , n} and x1 < x2 < x3 < . . . < xm. Set x0 = 0. Define a semigroup S(n, X) as follows: For any a, b ∈{1, 2, . . . , n} a ? b = { xi if xi−1 < a ≤ xi and b ≥ xm, max{a, b} if a > xm or b > xm. The author than shows that any rational commuting probability can be achieved by an appropriate choice of parameters. Contrastingly, for groups, Hegarty [18] showed that for any limit point3 l of Sc(G), l ∈ (29, 1], there is no increasing sequence of numbers {an}⊂ Sc(G), such that l = limn→∞ an. Recently, Buckley and MacHale investigated relations between the commuting spectra of finite groups and rings. Comparing the structure of these two spectra for large probabilities, the authors formulated two conjectures, [4], page 9: Conjecture 1. Sc(R) ⊂ Sc(G). Conjecture 2. Sc(R) = Sc(Gnil) or Sc(R) = Sc(G (2) nil). This paper positively resolves the first conjecture and partially resolves the second one.4 2. Main results Theorem 2.1. Sc(R) ⊆ Sc(G (2) nil) ⊆ Sann(Rsa ∩R (2) nil). 3 x is a limit point of a set S if every neighborhood of x contains at least one point of S different from x itself. 4 The authors would like to thank Victor Bovdi for his interest in this paper. 11 M. Juráš, M. Ursul / J. Algebra Comb. Discrete Appl. 9(1) (2022) 9–15 In [5], the authors determined all values in Sc(R) that are ≥ 1132. These are 1, 7 16 , 11 27 , 25 64 , 11 32 , and 22k + 1 22k+1 for k = 1, 2, 3, . . . . Thus, 1 2 6∈ Sc(R). But, 12 ∈ Sc(G), ([27], page 246), and so Sc(R) 6= Sc(G). In particular, Prc(S3) = 1 2 (see [20]); S3 denotes the symmetric group of order 3. This, together with the first inclusion of Theorem 2.1, positively resolves Conjecture 1. As for Conjecture 2, the Theorem states Sc(R) ⊆ Sc(G (2) nil). Now that we know Sc(R) is a subset of the potentially smaller one of the two sets, Sc(G (2) nil) and Sc(Gnil) (it is unknown whether or not Sc(G (2) nil) = Sc(Gnil)), we ask the following question: Does Sc(R) = Sc(G (2) nil) (2) hold true? We don’t know. But, Equation (2) does hold true, when restricted to finite groups and finite rings with odd number of elements. In fact, we prove the following: Theorem 2.2. Sc(ODD(R)) = Sc(ODD(G (2) nil)) = Sann(ODD(Rsa ∩R (2) nil)). Next, we would like to formulate a condition, purely in terms of probabilities in rings, that would imply Equation (2). Using Theorem 2.1, one obvious choice could be Sann(Rsa ∩ R (2) nil) ⊆ Sc(R). We can do slightly better. Because things are working smoothly when restricted to rings with odd number of elements, it is sufficient to focus on the "trouble makers" which are the 2-rings. Proposition 2.3. If Sann(Rsa ∩R (2) nil ∩R2) ⊆ Sc(R), then Equation (2) holds true. The condition of Proposition 2.3 implies a stronger statement: If Sann(Rsa ∩ R (2) nil ∩ R2) ⊆ Sc(R), then both inclusions in Theorem 2.1 can be replaced by equal signs. Note that if there is a counterexample to the condition above, i.e. if there exists a ring R such that R ∈ Rsa∩R (2) nil∩R2 and Prann(R) 6∈ Sc(R), then Prann(R) < 1132. We conjecture that Sc(R) = Sann(Rsa). 3. Proofs Let N be an associative nilpotent ring of class n. Then N, endowed with "circular multiplication", a ◦ b = a + b + ab, is a group which we will denote by GN.5 0 is the unit element in GN and a−1 = −a + a2 −a3 + · · ·+ (−1)n−1an−1 is the inverse of a in GN, a◦a−1 = a−1 ◦a = 0. Since, ab = ba if and only if a◦ b = b◦a, then, if N is finite, Prc(N) = Prc(GN), (3) Lemma 3.1. Let N be a nilpotent ring of class at most 3 (hence, also an associative ring). Let a, b, c ∈ N. Then (i) [a, b]GN = [a, b]N, (ii) [a, b]GN ◦ [c, d]GN = [a, b]N + [c, d]N, (iii) GN is a nilpotent group of class ≤ 2. 5 Another way to associate a group to a ring such that their commuting probabilities equate can be obtained by modifying a construction of Mal’cev [25]. For an arbitrary ring R, define a binary operation on R × R by (a, b) · (c, d) = (a + c, ac + b + d). This operation is associative, has unit (0, 0) and (a, b)−1 = (−a, a2 − b). G = (R × R, · ) is a nilpotent group of class at most 2 and Prc(R) = Prc(G). Note that, unlike the construction of GN, the ring R is not required to be nilpotent or associative! 12 M. Juráš, M. Ursul / J. Algebra Comb. Discrete Appl. 9(1) (2022) 9–15 Proof. (i) follows by direct computation. (ii). By (i), [a, b]GN ◦ [c, d]GN = [a, b]N ◦ [c, d]N = [a, b]N + [c, d]N + [a, b]N[c, d]N = [a, b]N + [c, d]N. (iii). By (i), [[a, b]GN , c]GN = [[a, b]N, c]N = 0. Let G be a nilpotent group of class ≤ 2 and let Z = Z(G) be the center of G. Then G/Z is abelian. By RG, denote the ring with the additive group G/Z ⊕Z, and the multiplication defined by (aZ, x) · (bZ, y) = (Z, [a, b]), (4) where [a, b] = a−1b−1ab is the commutator in G. Explicitly, the addition in RG is given by (aZ, x) + (bZ, y) = (abZ, xy). (Z, eG) is the zero element and (a−1Z, x−1) is the additive inverse of (aZ, x). To verify that RG is indeed a ring, the distributive laws have to be satisfied. Let a, b, c ∈ G and x, y, z ∈ Z. We have (cZ, z) · ((aZ, x) + (bZ, y)) = (cZ, z) · (abZ, xy) = (Z, [c, ab]). On the other hand, (cZ, z) · (aZ, x) + (cZ, z) · (bZ, y)) = (Z, [c, a]) + (Z, [c, b]) = (Z, [c, a][c, b]). Using [G, G] ⊆ Z, we deduce [c, a][c, b] = c−1a−1cac−1b−1cb = c−1[a, c−1]b−1cb = c−1b−1[a, c−1]cb = c−1b−1a−1cac−1cb = c−1b−1a−1cab = c−1(ab)−1c(ab) = [c, ab]. Hence, the left distributive law is satisfied. The proof of the right distributive law is similar.6 Lemma 3.2. Let G be a nilpotent group of class at most 2. Then RG is a strongly antisymmetric nilpotent ring of class at most 3. If G is finite, then |RG| = |G| and Prc(G) = Prann(RG). (5) Proof. |RG| = |G/Z||Z| = |G|. R3G = 0 and strong antisymmetry of RG follows immediately from the multiplication formula (4). To prove (5), it suffices to note that (aZ, x) · (bZ, y) = (Z, [a, b]) = (Z, eG) if and only if [a, b] = eG. But, this is exactly when ab = ba. Proof of Theorem 2.1. We first show that Sc(R) ⊆ Sc(G (2) nil). Let r ∈ Sc(R) By Lemma 1.1, there is a nilpotent ring N of class at most 3 such that r = Prc(N). By Lemma 3.1(iii), GN is a nilpotent group of class at most 2 and by Equation (3), Prc(GN) = Prc(N). We conclude that r ∈ Sc(G (2) nil). To prove the second inclusion, consider G ∈ G(2)nil. By Lemma 3.2 RG ∈ Rsa ∩R (2) nil and Prann(RG) = Prc(G). 6 Proposition 3 [4], states that the condition [c, a][c, b] = [c, ab] for all a, b, c ∈ G is equivalent to G being nilpotent of class ≤ 2. 13 M. Juráš, M. Ursul / J. Algebra Comb. Discrete Appl. 9(1) (2022) 9–15 Lemma 3.3. Let R be a finite antisymmetric ring and with odd number of elements. Then Prc(R) = Prann(R). Proof. In an antisymmetric ring, ab = −ba. Hence, ab = ba iff 2ab = 0. Since |R| is odd, 2ab = 0 iff ab = 0. Proof of Theorem 2.2. To prove Sc(ODD(R)) ⊆ Sc(ODD(G (2) nil)) ⊆ Sann(ODD(Rsa ∩R (2) nil)), we follow the proof of Theo- rem 2.1 and note that |N| = |GN| and |G| = |RG|. To conclude the proof of Theorem 2.2, it suffices to show Sann(ODD(Rsa ∩R (2) nil)) ⊆ Sc(ODD(R)). Let r ∈ Sann(ODD(Rsa ∩R (2) nil)) and let R ∈ ODD(Rsa ∩R (2) nil) such that r = Prann(R). By Lemma 3.3, r = Prann(R) = Prc(R) ∈ Sc(ODD(R)). For a noncommutative formal polynomial f(X, Y ) = aXY + bY X and rings R1 and R2, Prf (R1 ×R2) = Prf (R1) Prf (R2). (6) Let p be a prime number and let C be a class of finite rings. Denote Cp = C ∩ Rp. Assume C is closed under cartesian products. Then Cp is closed under cartesian products and both Sf (C) and Sf (Cp) are multiplicative monoids. Furthermore, values in Sf (C) are finite products of values taken from the set⋃ p Sf (Cp), where p runs over all prime numbers. We say that a class C of finite rings is hereditary, if any subring of a ring in C is also in R. If a class C is hereditary, then Cp is also hereditary. Proof of Proposition 2.3. It is easy to see that the class C = Rsa ∩ R (2) nil is hereditary and closed under cartesian products. Assume Sann(C ∩ R2) ⊆ Sc(R). By Theorem 2.2, Sann(ODD(C)) = Sc(ODD(R)) and so Sann(C ∩ Rp) ⊆ Sc(R) for a prime p 6= 2. Hence, for all primes p, the monoids Sann(C ∩ Rp) ⊆ Sc(R) and so Sann(C) ⊆ Sc(R). By Theorem 2.1, the reverse inclusion is satisfied, thus the proposition follows. 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. 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