What is the Probability that Two Group Elements Commute? Author(s): W. H. Gustafson Source: The American Mathematical Monthly, Vol. 80, No. 9 (Nov., 1973), pp. 1031-1034 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2318778 . Accessed: 29/09/2011 07:15 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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1973]

1031

MATHEMATICAL NOTES

This process worksbecause 13 = 22 x 3 + 1; 3613 = cause 13, 3613 are prime(cf. [1]). Noting that 79

=

22

x 3 x 7 x 43 + 1 and be-

2 x 3 x 13 + 1; 157 = 22 x 3 x 13 + 1; 547 = 2 x 3 x 7 x 13 + 1;

1093 = 22 x 3 x 7 x 13 + 1; 6709 = 22 x 3 x 13 x 43 + 1; 46,957 = 22 x 3 x 7 x 13 x 43 + 1; 303,493= 22 x 3 x 7 x 3613 + 1; 12,118,003= 2 x 3 x 13 x 43 x 3613 + 1; and that the numbersappearingon the lefthand side of the equations are prime (cf. [1]) we obtain the requiredresult. THEOREM

5. A lower bound on a numberN such that +(N)

one solutionis

=

M has exactly

1077.

Proof. Follows fromTheorems3 and 4. References 1. C. L. Baker,and F. J. Gruenberger, The first Six MillionPrimeNumbers,Santa Monica, The Rand Corporation, 1957. 2. Ivan Nivenand H. S. Zuckerman, An Introduction to theTheoryof Numbers, Wiley,New York, 1964. WHAT IS THE PROBABILITY THAT TWO GROUP ELEMENTS COMMUTE? W. H. GUSTAFSON,Indiana University

A studentstudyingbothprobabilityand algebramightwell ask 1. Introduction. the questionposed in the title.One can solve the problemfor finitegroupsby a straightforward attackas follows: Let G be a group of finiteorder n. The probabilityPr(G) that two elements selectedat random (with replacement)fromG are commutativeisf C In2, where C = {(x, y) E G x G jxy = yx} . In orderto count the elementsof C, we observe thatforeach x E G, thenumberofelementsof C of theform(x, y) is I Cx whereCx is the centralizerof x in G. Hence we have

JCj=

x lcxl,

wherethe sum extendsover all x E G. Now we recallthatif x and y are conjugate elementsof G, then Cx and Cy are conjugatesubgroups.Further,the numberof elementsin theconjugacyclass of x is [G: Cx]. Hence,ifx1,..., Xk are representatives of the conjugacyclasses in G, we have k

jCj= I

i =1

[G:CxJ]

ICxI

= k*n.

1032

W. H. GUSTAFSON

[November

Thus Pr(G) = k/n,the numberof classesin G dividedby the orderof G. This technique was used by Erdds and Turan [4]. Now let us observethat5/8is an upperbound.forPr(G) whenG is nonabelian. For the "class equation" tellsus that

I GI = jZj + I K,I +

+jKI,

whereZ is the centerof G, and K1, ..., K, are the nontrivialconjugacyclasses. We have I Kif ? 2 for i = 1, ...,t, whence ( G| -|Z|)/2 ? t. Thus k = t + IZ I|<

(IGj + fZ 1)/2.As G is nonabelian,G/Zis notcyclic(see Scott[7, p. 50]) andhence


IZ

2. Compact groups.The reader may now wonder whetherthe above analysis carriesover in any sense to infinitegroups. Of course,the ratio k/nis no longer but we shall see thatthereis an analogue of the bound 5/8fora class meaningful, of topologicalgroups. Let G be a compact,Hausdorfftopologicalgroup. We recall that G has a left Haar measure;thatis, a Borel measurep such that i(U) > 0 for each nonempty open set U of G, and u(x*E) = /(E) for each Borel set E of G and each x E G. Further,p is unique once we impose the normalizationconditionp(G) = 1. The readerwho is not familiarwithHaar measuremay consult[5, ChapterXI]. On the product space G x G, we impose the product measure p x p. Again let C = {(x,y)e G x GI xy = yx}. We remarkthat C = f-'(1), wheref: G x G -+ G is the continuousfunctiongivenbyf(x, y) = xyx- ly -. It followsthatC is closed, and hence measurable.We view p x p as a probabilitymeasure; then Pr(G) = p x p(C). Let us now prove our generalizationof the last resultof Section 1: THEOREM.

Let G be a compactnonabelian group. Then Pr(G) < 5/8.

Proof. Let X: G x G

-*

functionof C. Then we have reals,be the characteristic

/ux 4C)

=

TX d(G GXG

x u).

By Fubini's theorem[5, p. 148], we have /x

i(C) =

x(x,y)d1u(y)d1u(x).

of x in G. Also, fG X(X, y)d1i(y)= it(Cx)foreach x, whereagain Cx is thecentralizer We recallonce morethat[G: Z] ? 4. As G is thedisjointunion of thecosetsof Z, it followsthat1(Z) ? 1/4. (Note thatZ is closed and hence measurable.)Now we noticethatif x E Z, thenCx = G and so t(Cx) = 1; on theotherhand,ifx E G-Z, then Cx has index at least 2 in G, whencei(Cx) < 1/2. Thereforewe have

1973]

MATHEMATICAL NOTES

Pr(G)

=

=

i x u(C) =

f

f

1033

m(CQ)dm(x)

ji(Cx)dy(x)+ fGZ(cx)dy(x)

?< ,(Z) - 1 + p(G-Z) *1/2= p(Z) + 1/2- i(Z)/2 < 5/8.

3. Furtherremarks.Let us now returnto the case of finitegroups.Here the formulaPr(G) = k/nmaybe used to good advantagein calculatingboundson Pr(G) for special classes of groups. For example,the readermay use the class formula to showthatfornonabelianp-groupsG, Pr(G) < (p2 + p -_ )/p3. Some information may also be gatheredfromthe theoryof group characters[2]. One makes use of the factthatthe numberof irreduciblecomplexcharactersof G is just k, together [G: G'] + n2 + + n , where G' is the commutator with the fact that IGI subgroupof G, and n1,* nsare thedegreesof thenonlinearirreduciblecharacters. Here are some problemsforthe readerto try: (i) Pr(G x H) = Pr(G) *Pr(H).

(ii) If Pr(G)= 5/8,thenG is nilpotent. (iii) If G is finiteand Pr(G)= 5/8,thenG is thedirectproductof an abeliangroupand a 2groupH suchthatIH > 8, H is directly indecomposable and Pr(H)= 5/8. (iv) Characterize thegroupsH havingtheproperties in (iii). (See Miller[6],wherethegroups with[G:Z] = 4 areclassified.) (v) DerivetheboundPr(G) < 5/8forfinitegroupsby use of thefactsfromcharacter theory givenabove. (vi) If G is simpleand nonabelian, thenPr(G) < 1/12,withequalityforthealternating group on fiveletters. (Thisproblemwasfirst posedbyJ.Dixon.) (vii) Studytheprobabilistic properties of finitegroupsin general.Some starting pointsmight be Erd6sandTuran[4]and Dixon [3].Ofparticular interest is a conjecture ofDixon: The probabilitythattwoelements chosenat randomfroma finite simplegroupG generateG tends uniformly to one as theorderof G tendsto infinity.

Finally,we would like to encouragethe studyof a moredifficult problem:find lowerboundsforPr(G). Whileit is easy to see thatno universallowerbound exists, Erd6sand Turfan[4] haveshownthatPr(G)_ (log210g2 GI)/IGI. C. Ayoub[1] has developedsome lower bounds for p-groupsof small order. I am pleasedto acknowledge usefulconversations withM. Zorn,P. Halmos,and W. Moran. I am also grateful to R. MacKenziewhoreadtheoriginalmanuscript. References 1. C. Ayoub,On thenumberof conjugateclassesin a group,Proc.Internat. Conf.Theoryof Groups,GordonandBreach,New York,1967,pp. 7-10. 2. C. Curtisand I. Reiner,Representation Theoryof FiniteGroupsand Associative Algebras, Interscience, New York, 1962.

1034

[November

C. W. BARNES

group,Math.Z., 110( 1969)199-205. thesymmetric ofgenerating 3. J.Dixon,The probability IV, ActaMath.Acad. group-theory, ofa statistical 4. P. Erd6sand P. Turan,On someproblems Sci. Hung., 19 (1968) 413-435. N. J. 1950. 5. P. Halmos,MeasureTheory,Van Nostrand,Princeton, operatorsin a group,Proc.Nat. Acad. Sci. 6. G.A. Miller,Relativenumberof non-invariant USA, 30 (1944) 25-28. N. J. 1964. EnglewoodCliffs, 7. W. Scott,GroupTheory,Prentice-Hall,

REMARKS ON THE BESSEL

POLYNOMIALS

of Mississippi C. W. BARNES, University

The Besselpolynomialy"(x) is definedby Krall and Frink [4] 1. Introduction. to be the polynomialof degreen, withconstanttermequal to unity,whichsatisfies the differential equation (1)

x2y" + 2(x + 1)y' -n(

+ 1)y = 0.

Krall and Frinkdiscussedthe Bessel polynomialsfromthe standpointof recurrence relations,orthogonality, generatingfunctions,and related matters.Their algebraicpropertieswereconsideredby Grosswald[2]. In the presentnote we establisha new resultconcerningthe zeros of the Bessel polynomials.Using a test of Wall [7], whichthe Bessel polynomialsfitin a very naturalway,we provethattheirzeros have negativereal parts.We also give a new proof of a theoremof Dickinson [1], section6, thatthe originis a limitpoint of zeros of the Bessel polynomials.Our proof of Dickinson's theoremis somewhat simplerthanthatgivenin [1] inasmuchas it dependsmainlyon an applicationof the maximummodulus principlefor analyticfunctions. Finally we commenton the historyof the Bessel polynomials,and relatethem to workofOlds [5] based on Hermite[3]. The differential 2. The zeros of theBessel polynomials. equation (1) is satisfied by (2)

Yn(X) -

(I

k=O

(nifk)!

X k

21

(n-k)!k!

relations These polynomialssatisfythe recurrence (3)

Yn+l(X) = (2n + 1)xYn(x)+ Yn_l(x),X

where y0(x) = 1, y1(x) = 1 + x.

Krall and Frink [4] showedthat (4)

x2y'(x)

=

(nx - 1)Yn(x)+

l(x) Yn_