What\u27s the probability that two elements in a finite group commute? A formal answer,
Pr2(G) = {(x, y) [element of] G2 |xy = yx}| / |G|2
begs our next question. How many ordered pairs of elements of a finite group commute

Sherman, G. J.

Walker, Judy Leavitt

Walker, Mark E.

English

DigitalCommons@University of Nebraska

University of Nebraska - LincolnDigitalCommons@University of Nebraska - LincolnFaculty Publications, Department of Mathematics Mathematics, Department of1992Rewriteability in Finite GroupsJudy Leavitt WalkerUniversity of Nebraska - Lincoln, judy.walker@unl.eduG. J. ShermanRose-Hulman Institute of TechnologyMark E. WalkerUniversity of Nebraska - Lincoln, mark.walker@unl.eduFollow this and additional works at: https://digitalcommons.unl.edu/mathfacpubPart of the Applied Mathematics Commons, and the Mathematics CommonsThis Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. Ithas been accepted for inclusion in Faculty Publications, Department of Mathematics by an authorized administrator of DigitalCommons@Universityof Nebraska - Lincoln.Walker, Judy Leavitt; Sherman, G. J.; and Walker, Mark E., "Rewriteability in Finite Groups" (1992). Faculty Publications, Department ofMathematics. 185.https://digitalcommons.unl.edu/mathfacpub/185 Rewriteability in Finite Groups J. L. Leavitt, G. J. Sherman and M. E. Walker INTRODUCTION. What's the probability that two elements in a finite group commute? A formal answer, Pr2(G) = {(X, y) E G2Ixy = yx) I 2 I~~~G 12'(1 begs our next question. How many ordered pairs of elements of a finite group commute? Let's be specific. Consider the "commutativity matrix" for the symmetric group on three symbols. S3 id (1,2) (1,3) (2,3) (1,2,3) (1,3,2) id 1 1 1 1 1 1 (1,2) 1 1 0 0 0 0 (1,3) 1 0 1 0 0 0 (2,3) 1 0 0 1 0 0 (1,2,3) 1 0 0 0 1 1 (1,3,2) 1 0 0 0 1 1 The xth row of this matrix identifies the subgroup, C(x), of elements which commute with x; i.e., the centralizer of x. Here's the way to parse the commutativ- ity count for S3. 18 = 6 + 2 + 2 + 2 + 3 + 3 = 1 * 6 + 3 * 2 + 2 3 = 6 + 6 + 6 = 3 6 The elementary group theory at work in this count is: * conjugate elements have centralizers of the same order y = g- xg implies C(y) = g-1C(x)g, * the order of a conjugacy class is the index of the centralizer of any element in the class IxGI =j{g-lxglg E G) I [G: C(x)], * Lagrange's theorem IGI = [G:H] IHI. 446 REWRITEAA- ILITY IN FINITE GROUPS [May An abstraction of this example, originally due to Erdos and Turfan [4], answers our second question. {(x,y) E G2Ixy yx = C(x) xeG k = IIxGI |C(Xi)l i=1 k = [G:C(xi)] iC(xi)I i=1 =k* IGI (2) where {x1, x2, ... , Xk} is a complete set of conjugacy class representatives of G. Thus, an informative answer to our first question is k Pr2(G) = IGI It comes as no surprise that G is abelian precisely when Pr2(G) = 1. But what may surprise you is that if G is not abelian, then k p2 +Ps -l 5 Pr(G= _ <3- 2 ~IGI < (3)8 where ps is the smallest prime divisor of the order of G. The essence of these bounds is that the index of the center of a nonabelian group is at least p2; i.e., IG:Z] p2 The 5/8 bound, which is assumed by the dihedral and quaternion groups of order eight, has been around for a long time. Yet, it doesn't seem to be commonly known-so be sure to tell your students about it. We do not know with whom it originated. Some say Max Zorn. But, many years ago, during a conversation with one of the authors (Sherman), Zorn declined credit for the bound. To the best of our knowledge the bound first appeared in print in 1973 when Gustafson [7] showed that an analogous bound holds for compact nonabelian groups. Gallian's recent textbook ([6, pages 329, 330]) also includes a discussion of the bound. Both upper and lower bounds on Pr2(G) for various classes of groups have been obtained ([1], [4], [5], [7], [10], [13]). And, since commutativity can be defined in terms of conjugation, analogous results have been pursued for various group actions ([11], [13], [15]). Commutativity is a special case of rewriteability. Let S c Sn - {id}; i.e., S is a set of nontrivial permutations of {1, 2, ... n}. An n-tuple (x1, x2, .. ., xn) of ele- ments of G is S-rewriteable if x1x2 ... = xn (1) x, (2) * xou(n) for some a E S. We generalize (1) by setting Prn(G; S) |Rwn(G; S)| (4) I GIn where Rwn(G; S) = {(x1 x2 ...xn) E Gnl(xl x2 ... xn) is S-rewriteable). (5) Those groups for which Prn(G; Sn - {id}) = 1 will be referred to as n-rewriteable groups. The notion of rewriteability has its origins in automata theory and is currently of considerable interest in group theory [2]. 1992] REWRITEABILITY IN FINITE GROUPS 447 In particular, Curzio, Longobardo and Maj [3] have provided elementary proofs that the following three statements are equivalent. i) G is 3-rewriteable; i.e., xyz E {yxz, zyx, xzy, zxy, yzxl for all x, y, z E G. ii) The order of the derived subgroup of G, G' = x- 1y -1xy Ix, y E G> is one or two. iii) The order of the centralizer of each element of G is I G I or I G 1 /2. The equivalence of ii) and iii) revolves around the relationship between commu- tators (elements of the form x-ly-1xy) and conjugates: x-ly-lxy = g if, and only if y -xy = xg. The equivalence of i) with ii) or iii) is case-driven. For example, an application of the definition of 3-rewriteability to the product Xyx2 places x2 in the center of the group. This means the centralizer of x is a "large" normal subgroup. In view of iii) and our discussion prior to (2), we may add the following statement to the list. iv) The order of each conjugacy class of G is one or two. Each of ii), iii) and iv) suggests a connection between 3-rewriteability and the probability of two elements commuting. In particular, the size of a group's derived subgroup is a classic measure of the degree of commutativity the group enjoys. If G' is small, then "most" commutators are trivial; i.e., it is "likely" that xy = yx. Let's formalize this connection. Notice that the average order of a conjugacy class of a 3-rewriteable group is less than two; i.e., I G I/k < 2. Thus Pr2(G) = k/lIGI > 1/2 for 3-rewriteable groups. An appeal to character theory establishes the converse. G has k irreducible characters and I G I/ IG' irreducible characters of degree one. Thus IGI 2 (IGI/IG'I) . 12 + (k - IGI/IG'I) . 22 which implies 1 ? -3/IG'I + 4k/lGl. If k/IGI > 1/2, then 1 > -3/IG'I + 2 from which it follows that IG'I < 2. We have the following theorem. Theorem. A finite group G is 3-rewriteable if, and only if, Pr2(G) > 1/2. It's interesting to formulate this theorem in terms conjugacy classes Each conjugacy class has order one or two if, and only if, the average conjugacy class order is less thah two. and in terms of conditional probability. The probability of x and y commuting, given y, is at least 1/2 for each y, if and only if Pr2(G) > 1/2. AN ELEMENTARY PROOF. An elementary proof that if Pr2(G) > 1/2, then G is 3-rewriteable follows. Think of "3-rewriteable" as a generic label for your favorite from among statements i)-iv) above. We will assume that G is not 3-rewriteable and prove that Pr2(G) < 1/2. 448 REWRITEABILITY IN FINITE GROUPS [May The proof and subsequent discussion hinge on relationships among the orders of three subsets of G: X = {x E GI[G: C(x)] 2 3), Y = {x E GI[G: C(x)] = 2), Z = {x E GI [G: C(x)] = 1}; i.e., the center of G. The following three lemmas, which are of some interest in their own right, help organize the proof. Lemma 1. If x and y are elements of G for which [G: C(x)] = 2 and C(y) n (G - C(x)) # 0, then [G: C(xy)] 2 [G: C(y)]. Proof: The conjugacy class of y in G, yG, may be written {ygl, y92,. .., ygn} where {g1, g2, .. ., gn} is a complete set of right coset representatives for C(y) in G. Moreover, we may choose each coset representative in C(x). Otherwise C(y)gi c G - C(x), which means that G - C(x) = C(x)gi since [G: C(x)] = 2. Therefore C(y)gi c C(x)gi and so C(y) c C(x), a contradiction. The conclusion follows because the mapping y g -- xy gi embeds yG in (xy)G. Lemma 2. If at least 3 *Z I elements of G have centralizers of index at least 3, then Pr2(G) < 1/2. Proof Observe that IRw2(G)I = k * IGI < (IXI/3 + IYI/2 + IZI) * IGI = (lZI + (IXI - 3 X ZI)13 + 1I1l2 + IZI) *IGI < (IZI + (IXI - 3 IZI)/2 + IYI/2 + IZI) IGI = (IXI + IYI + IZI) * IGI/2 = IG12/2. Thus Pr2(G) < 1/2 as claimed. Lemma 3. If G is not 3-rewriteable, then [G: Z] ? 6. Proof If [G: Z] is 1, 2, 3 or 5, then G is abelian since G/Z is cyclic. If [G: Z] = 4 and x is a non-central element, then Z c C(x) c G implies [G: C(x)] - 2; i.e., G is 3-rewriteable. It isn't necessary to invoke the centralizer characterization of 3-rewriteability to complete the proof of Lemma 3. If [G: Z] = 4, then G/Z - Z2 E Z2. Thus G = Z U xZ U yZ U xyZ. The only triple products from G whose 3-rewriteability we might question have form (xz1)(yz2)(xyz3) or (xz1)(xyz2)(yz3). But, notice that (Xz1)(YZ2)(XYZ3) = (xyz3)(xz1)(yz2) and that (xz1)(xyz2)(yz3) = (yz3)(xz1)(xyz2) because x2 E Z. This proof makes Lemma 3, which is an analogue of the fact that [G: Z] ? 4 for nonabelian G, an appealing student exercise. Now we can weave that elementary proof we promised. Note that X # 0 since G is not 3-rewriteable. Choose g E X and set n = [G: C(g)]. Then Z U Zg c C(g) and (Z U Zg) n Y = 0. Thus I C(g) n YI < I G I /n - 2 IZ I and so I (G - C(g)) n YI ? IYI - IGIl/n + 2 > IZI. If x c (G - C(g)) n Y, then [G: C(x)] = 2 and C(g) n (G - C(x)) # 0 implies, by Lemma 1, that [G: C(xg)] ? [G: C(g)] ? 3. 1992] REWRITEABILITY IN FINITE GROUPS 449 Therefore (G - C(g)) n Y c X; in fact (G - C(g)) n Y c X - Zg as Zg c X n C(g). Thus IXI- IZI = IX-Zgl > I(G-C(g)) n Yl 2 IYl - lGl/n + 2 - IZI; i.e., IXI 2 iYI - lGi/n + 3 - IZI. (6) In view of Lemma 2 and (6) we are done if I YI 2 I G l /3, so assume I Yl < I G I /3. In this case Lemma 3 implies that IXl > I GI /2 and, therefore, that IXl > 3 I Z I. The theorem is proved. Corollary. If G is not 3-rewriteable, then at least I G I * (n - 1)/2n + I Z I elements of G have centralizers of index at least 3 where n is the greatest centralizer index among the elements of G. In particular, more than 1/3 of the elements of G have centralizers of index at least 3. Proof: This follows directly from (6) by substituting I G I - IXI for I Yl + I Z 1. The 1/2 bound for 3-rewriteability is sharp in two senses. i) Pr2(G) = 1/2 if, and only if, G/Z S S3. Our opening example suggests the involvement of S3. That Pr2(G) = 1/2 implies G/Z - S3 is a straight forward application of Lemma 3 and the Corollary. The converse follows since IXi = 3 I Z I and lYl = 2 - IZI for groups satisfying G/Z S3. ii) There exists a sequence, {Gn}, of 3-rewriteable groups such that Pr2(Gn) J, 1/2. But where? A result of Ito [9] says that groups in which each conjugacy class is of order one or p, for a fixed prime p, must be the direct product of a p-group (a group whose order is a power of p) with this property and an abelian group. Thus, if G is 3-rewriteable we may write G _ T x A, where T is a 3-rewriteable 2-group and A is abelian. Conjugacy classes in direct products are direct products of conjugacy classes, so Pr2(G) = Pr2(T X A) = Pr2(T) * Pr(A) = Pr2(T). Net result: we may restrict our attention to 2-groups. The quaternion group of order eight, mentioned in conjunction with the 5/8 bound, is worth a look: Q =X y, zlx2 =y2 = z2 = x-1y- xy = x-lz-lxz = e, y- z-'yz = x>. The relevant facts are; IQI = 8 = 2 z = Q= {e,x}), k = 5 = IZI + (IGI- IZI)/2 = (IGI + IZI)/2, Pr2(Q) = 5/8 = 1/2 + lZl/(2 * IGI). We generalize by taking Gn to be (an extra-special 2-group [12]) generated by xl, x2,..., x2n+1 subject to the relations X2 = e for 1 < i < 2n + 1, -z 1x 1xx.= (x1 forievenandj=i + 1, xi xi xix=  e otherwise. Then IGnl = 22n+1 and Z = G' = {e, x1} so that Pr2(Gn) = k/IGnl = 1/2 + 1/22n+1 A PROBLEM. We encourage study of the problem of determining bounds for Prn(G; S). The following lemma generalizes (3) and prompts a conjecture. 450 REWRITEABILITY IN FINITE GROUPS [May Lemma 4. If n 2 2 and a E Sn - {id}, then IRwn(G;{r))DI < k - IGI'n . Proof: The proof is by induction on n. The case for n = 2 was made in (2). Now assume the result holds for n - 1. If o(n) = n, then x1x2 ... xn = x.(.)xa(2) ... xo(n) if, and only if, x1x2 ... Xn-1 = X(1)X(2) ... X(n - 1)' Therefore IRwn(G; {oa})I = IRwn_1(G; {})I - IGI where a' is a restricted to {1, 2, ... , n - 1). The induction hypothesis yields the result. If o(n) < n, say a(n) = m, then xlx2 ... xn = Xo.(1)Xo(2)* X u(n) if, and only if, XlX-I) *Xc(l)X * = XU(I+l)X1(I+2) * Xm where a(j) = n. Let g = x-. Xj)X...... x,_1 and h= *-- Tm. Notice that I{x Ixn-gxn = h}I is IC(g)I or 0 for fixed x1, x2' ... Xn-1i and that g varies over G as xm varies over G. Thus IRwn(G;{a))l < EJ ... E EC(g)j X1 Xm Xn-1 =E, * .. E (LC(g) I X1 Xn-1 xm =E ... * * I C(9) Xi Xn-1 g =E .. E (k IGI) Xl Xn-1 =kIG In-1 as claimed. It follows from (3) and Lemma 4 that Prn(G;S) =|Rwn(G;S)|/IGI I SI * k/IGI = ISI Pr2(G) < ISI (p2 +p5 - 1)/p3 (7) Since (p2 + PS - 1)/p 3 J 0 as p5 -*oo we may use (7) to conclude that, for 1SI fixed and sufficiently large ps, a "5/8-like" bound exists for Prn(G; 5). Random sampling (using CAYLEY [8]) of the "S-rewriteability hypercube" of various groups suggests such bounds exist independent of p5. Conjecture. If G is not S-rewriteable then there exists Pn(S) < 1, independent of G, such that Prn(G; S) < Pn(S) < 1. Specifically, if p5 2 7, then Pr3(G; S3 - {id}) < 275/343. However, CAYLEY suggests Pr3(G; 53 - {id}) < 17/18. Thus for 3-rewriteability our conjecture is: If G is not 3-rewriteable, then Pr3(G; 53 - {id}) < p3(3 - {id}) = 17/18. If this conjecture proves to be true, then the 17/18 bound is sharp because Pr3(S3; 53 - {id}) = 17/18. We conclude by observing that if G is a non-abelian finite simple group then Pr3(G, 53 - {id}) < 5/12. This follows from (7) because Pr2(G) < Pr2(A5) [5] and Pr2(A5) = 1/12. It seems likely that the bound is actually 27/100 because CAYLEY shows Pr3(A5, S3 - {id}) to be 27/100. ACKNOWLEDGMENT. The authors thank the referees for their suggestions. 1992] REWRITEABILITY IN FINITE GROUPS 451 REFERENCES 1. E. A. Bertram, A density theorem on the number of conjugacy classes in finite groups, Pacific J. Math., 55 (1974) 329-333. 2. R. D. Blyth and D. J. S. Robinson, Recent progress on rewriteability in groups, J. London Math. Soc., to appear. 3. M. Curzio, P. Longobardi and M. Maj, Su di un problema combinatorio in teoria dei gruppi, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur, (8) 74 (1983), 136-142. 4. P. Erdos and P. Turan, On some problems of a statistical group-theory, IV, Acta Math. Acad. Sci. Hung., 19 (1968) 413-435. 5. J. D. Dixon, Problem 176, Canad. Math. Bull., 16 (1973) 302. 6. J. A. Gallian, Contemporary Abstract Algebra, 2nd edition, D. C. Heath, 1990. 7. W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly, 80 (1973) 1031-1034. 8. D. F. Holt, The CAYLEY group theory system, Notices of the American Mathematical Society, 35 (1988). No. 8, 1135-1140. 9. N. Ito, On finite groups with given conjugate types, Nagita Math. J., 6 (1953), 17-28. 10. E. Landau, Klassenzahl binarer quadratischer Formen von negativer Discriminante, Math. Annalen, 56 (1902) 671-676. 11. T. J. Laffey and D. MacHale, Automorphism orbits of finite groups, J. Austral. Math. Soc. (Series A), 40 (1986) 253-260. 12. D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, 1982. 13. G. J. Sherman, A probabilistic estimate of invariance for groups, Amer. Math. Monthly 85 (1978) 361-363. 14. , A lower bound for the number of conjugacy classes in a finite nilpotent group, Pacific J. Math., 79 (1979) 253-254. 15. G. J. Sherman, T. J. Tucker and M. E. Walker, How Hamiltonian can a finite group be? Archives Math., (Basel), to appear. University of Michigan Ann Arbor, MI 48109 Rose-Hulman Institute of Technology Terre Haute, IN 47803 New Mexico State University Las Cruces, NM 88003 The beginning of wisdom is the defini- tion of terms. -Socrates (470?-399 B.C.) 452 REWRITEABILITY IN FINITE GROUPS [May

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