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

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Faculty Publications, Department of Mathematics Mathematics, Department of
1992
Rewriteability in Finite Groups
Judy Leavitt Walker
University of Nebraska - Lincoln, judy.walker@unl.edu
G. J. Sherman
Rose-Hulman Institute of Technology
Mark E. Walker
University of Nebraska - Lincoln, mark.walker@unl.edu
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Walker, Judy Leavitt; Sherman, G. J.; and Walker, Mark E., "Rewriteability in Finite Groups" (1992). Faculty Publications, Department of
Mathematics. 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
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 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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