We examine the total possible number of commutativities in a finite algebraic system, concentrating on groups, but also examining rings and semigroups. Numerical restrictions are found and bounds for the total number of commutativities in subgroups and factor groups are derived. Finally, a curious connection with group representations is explored

Dolan, Brian

MacHale, Desmond

MacHale, Peter

Cork Open Research Archive

UCC Library and UCC researchers have made this item openly available.Please let us know how this has helped you. Thanks!Title Counting commutativities in finite algebraic systemsAuthor(s) Dolan, Brian; MacHale, Desmond; MacHale, PeterPublication date 2016Original citation Dolan, B., MacHale, D. and MacHale, P. (2016) 'CountingCommutativities in Finite Algebraic Systems', Bulletin of the IrishMathematical Society, 77, pp. 61-70.Type of publication Article (peer-reviewed)Link to publisher'sversionhttp://banach.ucd.ie/bull77/index.phpAccess to the full text of the published version may require asubscription.Rights © 2016 Irish Mathematical SocietyItem downloadedfromhttp://hdl.handle.net/10468/9600Downloaded on 2021-11-27T09:33:48ZIrish Math. Soc. BulletinNumber 77, Summer 2016, 61–70ISSN 0791-5578Counting Commutativities in Finite Algebraic SystemsBRIAN DOLAN, DES MACHALE AND PETER MACHALEAbstract. We examine the total possible number of commutativ-ities in a finite algebraic system, concentrating on groups, but alsoexamining rings and semigroups. Numerical restrictions are foundand bounds for the total number of commutativities in subgroupsand factor groups are derived. Finally, a curious connection withgroup representations is explored.1. IntroductionConsider the Cayley table of a finite group G. For a, b,∈ G, ifab = ba, we place a 1 in each of the boxes corresponding to ab andba. This is called a commutativity in G. Otherwise we put a 0 ineach of these boxes, indicating a non-commutativity in G. If G isan abelian group, there will be a 1 in each box, so we disregard thisuninteresting case.We call this matrix of 1’s and 0’s the commutativity chart for G.Here for example is the commutativity chart for S3, the group of allpermutations on the set {1, 2, 3} under composition. S3 is in factthe smallest non-abelian group.e (123) (132) (12) (13) (23)e 1 1 1 1 1 1(123) 1 1 1 0 0 0(132) 1 1 1 0 0 0(12) 1 0 0 1 0 0(13) 1 0 0 0 1 0(23) 1 0 0 0 0 1We denote by I(G) the number of times that 1 appears in the com-mutativity chart and by O(G) the number of times that 0 appears.Thus I(S3) = 18 and O(S3) = 18 also.In general we see that I(G) + O(G) = |G|2 and O(G) > 0 sincewe are assuming G is non-abelian. Also we have I(G) > 0 since for2010 Mathematics Subject Classification. 20F99.Key words and phrases. Commutativities, Groups.Received on 29-3-2016.c©2016 Irish Mathematical Society6162 DOLAN, MACHALE AND MACHALEexample xx = xx for all x ∈ G. One of our objectives of this notewill be to discuss the possible values of I(G) and O(G), where G isa finite non-abelian group and to investigate the values of I(S) andO(S) for other non-commutative algebraic systems S.Since if ab 6= ba then ba 6= ab and xx = xx for all x, we see thatO(G) is always an even number, but there are examples to show thatI(G) can be either even or odd. For example, I(A4) = 48, where A4is the alternating group of order 12, while I(G(21)) = 105, whereG(21) is the non-abelian group of order 21. We emphasise thatthroughout, G denotes a finite non-abelian group.2. Some Elementary ResultsLet us recall some facts from elementary group theory. Two ele-ments x and y in G are said to be conjugate if there exists w ∈ Gwith y = w−1xw. The relation of conjugacy is easily seen to bean equivalence relation on G, under which G is partitioned into dis-joint conjugacy classes. For example, in the group S3, the conjugacyclasses are {e}, {(123), (132)} and {(12), (13), (23)}.In general, let G have exactly k(G) conjugacy classes and let Cl(x)be the class containing x. Let CG(x), the centralizer of x in G, bethe subgroup of G given by CG(x) = {a ∈ G | ax = xa}. Thereis a nice connection between conjugacy classes and centralizers viz.|Cl(x)| = (G : CG(x)), i.e. the number of cosets of CG(x) in G, andboth these numbers are divisors of |G|.From the definition, we have thatI(G) =∑x∈G|CG(x)| =∑x∈G|G||Cl(x)|= |G|∑x∈G1|Cl(x)|= |G|k(G). See [5].It follows that O(G) = |G|2 − I(G) = |G|(|G| − k(G)). Thusin the case of S3, since k(S3) = 3, we have I(G) = 6 · 3 = 18 andO(G) = 6 ·(6−3) = 18, in agreement with our previous calculations.Theorem 2.1. If |G| is odd, then k(G) is odd.Proof. If |G| is odd, since O(G) is even and O(G) = |G|(|G|−k(G)),we see that |G| − k(G) must be even, so k(G) is odd. We note that the converse of this result is not true; k(S3) = 3, but|S3| = 6.Counting Commutativities 63Theorem 2.2. I(G) is odd if and only if |G| is odd.Proof. If |G| is odd then by Theorem 2.1 k(G) is odd, so I(G) =|G|k(G) is odd. Conversely, if I(G) is odd then |G| clearly must beodd. In fact the smallest possible odd value of I(G) = 105 = 21 ·5, aris-ing from G(21), which is the smallest odd-order non-abelian group.We remark that Theorem 2.1, which says that if |G| is odd, then|G| − k(G) ≡ 0 (mod 2), can be improved upon considerably us-ing the theory of matrix group representations. A lovely theorem ofBurnside [3] states that if |G| is odd, then |G|−k(G) ≡ 0 (mod 16).Again G(21) shows that this result is the best possible. SinceO(G) = |G|(|G| − k(G)) we haveTheorem 2.3. If |G| is odd, then O(G) is a multiple of 16|G|.Again, O(G(21)) = 336 = 16 ·21, shows that this result is the bestpossible.We now investigate the possible values of I(G) and O(G) as Granges over all finite non-abelian groups. For a given group G it iseasy, if tedious, to calculate the value of k(G), and for certain classesof groups, and for groups of small order, this information is readilyavailable from a variety of sources.In particular let Dn be the dihedral group of order 2n (n > 2)given by< a, b | an = 1 = b2; b−1ab = a−1 >Then if n(= 2m) is even, we have k(D2m) = m+3, making I(D2m) =4m(m + 3) = 4m2 + 12m.If n(= 2m + 1) is odd, then k(D2m+1) = m + 2, so I(D2m+1) =(4m + 2)(m + 2) = 4m2 + 10m + 4.The values of O(Dn) can be found from O(G) = |G|2 − I(G).The symmetric group Sn of order n! has exactly p(n) conjugacyclasses, where p(n) is the (integer) partition function, so I(Sn) =n!p(n) and O(Sn) = n!(n!− p(n)).For distinct odd primes p and q, with p < q where p|(q − 1),there is a unique non-abelian group G(pq) of order pq. Easy cal-culations show that G(pq) has exactly p + q−1p conjugacy classes,so that I(G(pq)) = q(p2 + q − 1) and O(G(pq)) = p2q2 − I(G) =q(q − 1)(p2 − 1).We now present a chart with three columns. In the first column arethe possible orders of a finite non-abelian group G. In the second64 DOLAN, MACHALE AND MACHALEand third columns we give the values of I(G) and O(G) for eachnon-abelian group of order |G|. Since it is known that there areonly finitely many groups with a given order and also only finitelymany groups with a given number of conjugacy classes ([6], [9]) ,we see that there are just finitely many (maybe zero) groups with agiven I(G) or a given O(G). Note that there may be several differentgroups of order |G| with the same k(G) and hence the same I(G)and O(G).|G| I(G) O(G)6 18 188 40 2410 40 6012 48 9612 72 7214 70 12616 112 14416 160 9618 108 21618 162 16220 100 30020 160 24021 105 33622 154 33024 120 45624 168 40824 192 38424 216 36024 288 28824 360 21626 208 46827 297 43228 280 50430 270 63030 360 54030 450 45032 352 67232 448 576|G| I(G) O(G)32 544 48034 340 81636 216 108036 324 97236 360 93636 432 86436 648 64838 418 102639 273 124840 400 120040 520 108040 640 96040 1000 60042 294 147042 420 134442 504 126042 630 113442 882 88244 616 132046 598 151848 384 192048 480 182448 576 172848 672 163248 720 158448 768 153648 864 144048 1008 1296|G| I(G) O(G)48 1152 115248 1440 86450 700 180050 1000 150052 364 234052 832 187254 540 237654 810 210654 972 194454 1188 172854 1458 145855 385 264056 448 268856 952 218456 1120 201656 1960 117657 513 273658 928 243660 300 330060 540 306060 720 288060 900 270060 1080 252060 1200 240060 1440 216060 1800 1800We note that for direct products of groups G1 and G2, I(G1 ×G2) = I(G1)I(G2) and k(G1 ×G2) = k(G1)k(G2). However,O(S3)O(S3) = 18 · 18 = 324 6= 972 = O(S3 × S3).By [7] we have k(G)|G| ≤58 so I(G) ≤58 |G|2, and O(G) ≥ 38 |G|2.Also, by examining Cayley tables, it is clear that I(G) ≥ 3|G|−2,so that O(G) ≤ |G|2 − 3|G|+ 2.Counting Commutativities 65Thus, consulting the above charts, we see that the allowable valuesfor I(G) are: 18, 40, 48, 70, 72, 100, 105, 108, 112, 120, 154, 160,162, 168, 192, 208, 216, 270, 273, 280, 288, 294, 297, 300, 324, 340,352, 360, 364, 384, 385, 400, 418, 432,. . .Similarly the allowable values for O(G) are: 18, 24, 60, 72, 96,126, 144, 162, 216, 240, 288, 300, 330, 336, 360, 384, 408, 432, 450,456, 468, 480, 504, 540, 576, 600, 630, 648, 672,. . .We mention that the function |G| − k(G) is examined in consid-erable detail in [1] . Also, one can show that I(G) = O(G) if andonly if G/Z(G) = S3, where Z(G) is the centre of G.3. Subgroups and Factor GroupsGallagher [4] gives elementary proofs of the following results forall finite groups G, where H is a subgroup of G and N is a normalsubgroup of G.(i) k(H) < (G : H)k(G), for H 6= G;(ii) k(G) ≤ (G : H)k(H);(iii) k(G) ≤ k(G/N)k(N).In our notation, these results immediately becomeTheorem 3.1. (i) I(H) < I(G) if H 6= G;(ii) I(G) ≤ (G : H)2I(H);(iii) I(G/N) ≥ I(G)/I(N).4. Other Algebraic SystemsLet S = {a, b} be a set of cardinality 2. Define a binary operation∗ on S as follows∗ a ba a bb a bEasy calculations show that S is a non-commutative semigroup withI(S) = 2 = O(S), so the sequences of allowable value of I(S) andO(S) for semigroups are different from those of I(G) and O(G) forgroups.The reader is invited to determine the sequences of allowable val-ues of I(S) and O(S) for non-commutative semigroups.Moving on to rings, consider the following set of 2 × 2 matricesover Z2 under matrix addition and multiplication mod 2:R = {( 0 00 0 ), ( 0 10 1 ), ( 1 01 0 ), ( 1 11 1 )}66 DOLAN, MACHALE AND MACHALEIt is easy to see that {R,+, ·} is a non-commutative ring of order 4.The commutativity chart for {R, ·} looks as follows:( 0 00 0 ) (0 10 1 ) (1 01 0 ) (1 11 1 )( 0 00 0 ) 1 1 1 1( 0 10 1 ) 1 1 0 0( 1 01 0 ) 1 0 1 0( 1 11 1 ) 1 0 0 1Thus I(R) = 10 and O(R) = 6. This single example shows that thesequences of allowable values of I(R) and O(R) for finite rings aredifferent from those for finite groups.Again the reader is invited to investigate this problem for otheralgebraic systems such as near–rings, loops, quasigroups etc.We remark that if S is a set with |S| = n we can always chooseclosed binary operations ∗ and ◦ on S such that I(S, ∗) = n (n > 1),and O(S, ◦) = 2n (n arbitrary).For example, if S = {a, b, c} define ∗ by∗ a b ca a a cb b b bc a c cto achieve I(S, ∗) = 3 and similarly for the general case.◦ a b ca a a ab b a ac b b cAlso in the second example O(S, ◦) = 6 and similarly for thegeneral case.5. A Connection with Matrix Representations ofGroupsThere is a surprising connection between I(G) and matrix rep-resentations of G. For definitions we refer the interested reader to[5].Let di, 1 ≤ i ≤ k, be the degrees of the irreducible complex matrixrepresentations of a finite group G i.e. the sizes of the square matri-ces involved. There are k(G) of these where G has k(G) conjugacyclasses.Counting Commutativities 67Let T (G) =k(G)∑i=1di.[For example, for S3, (d1, d2, d3) = (1, 1, 2) so T (S3) = 4.]Using the Cauchy-Schwarz inequality on (1, 1, 1, . . . , 1) and(d1, d2, d3, . . . , dk) as in [8] , and remembering thatk∑i=1d2i = |G|, wefind that(T (G))2 < k(G)|G| = I(G). (G non–abelian)Let us see how this inequality looks for some specific groups ofsmall order.[We use the notation Qn for the dicyclic group of order 4n for n > 1where Qn = 〈a, b|a2n = 1; b2 = an, b−1ab = a−1〉].Group (T (G))2 I(G)S3 16 18D4 36 40Q2 36 40 (quaternion group)D5 36 40D6 64 72Q3 64 72A4 36 48D7 64 70S4 100 120When we write T (G) <√I(G) in a particular case such as D4, weget T4 <√I(D4) =√40 = 6.3245. Now T (D4) is an integer soT (D4) ≤ 6 and 6 is actually the correct answer!Similarly in the case of S4, we get T (S4) <√120 = 10.95445.Again T (S4) is an integer, so T (S4) ≤ 10 which gives the correctvalue of T (S4) = 10.It is remarkable that such a basic function as I(G), whose valuescan be read from the Cayley table, can be used to find informationabout T (G), which would appear to be a much more advanced grouptheoretic concept.6. Analogues of I(G) and O(G)There are so many analogies between k(G) and T (G) (as definedin section 5) that we make the following definitions:68 DOLAN, MACHALE AND MACHALEFor a finite non–abelian group G, let N(G) = |G|T (G) and M(G) =|G|(|G| − T (G)).It is not immediately clear what the interpretations of N(G) andM(G) are, but these functions have many properties analogous toI(G) and O(G). To save space we state results only, but methodsof proof are very similar to those for results concerning I(G) andO(G). We remark that the properties of |G| − T (G) are examinedin some detail in [2] .Theorem 6.1. I(G) < N(G) and O(G) > M(G).Theorem 6.2. There are only finitely many groups G (maybe zero)with a given N(G) or a given M(G).Theorem 6.3. N(G) is odd if and only if |G| is odd.Theorem 6.4. If |G| is odd, M(G) is a multiple of 4|G|.Theorem 6.5. If H is a proper subgroup of G, then N(H) < N(G).Theorem 6.6. M(G) is always even.Theorem 6.7. N(G) < |G| 32 (k(G)) 12 .Theorem 6.8. N(G1 ×G2) = N(G1) ·N(G2).Theorem 6.9. For the non–abelian group G(pq), we have N(G) =pq(p+q−1) and M(G) = pq(p−1)(q−1), where p and q are distinctodd primes.Theorem 6.10. N(G) ≤ 34 |G|2 and M(G) ≥ 14 |G|2.Finally, we give a chart of values of N(G) and M(G) for non–abelian groups G of small order which leads to information aboutthe sequences of allowable values of N(G) and M(G).Counting Commutativities 69|G| N(G) M(G)6 24 128 48 1610 60 4012 72 7212 96 4814 112 8416 120 13616 192 6418 120 20420 160 24020 240 16021 189 252|G| N(G) M(G)22 264 22024 240 33624 288 28824 336 24024 384 29224 432 14426 364 31227 405 32428 448 33630 480 42030 540 36030 600 300The sequence of allowable values of N(G) thus begins 24, 48, 60, 72, 96, 112,120, 160, 189, 192, 240, 264, 288, . . .The sequence of allowable values of M(G) thus begins 12, 16, 40, 48, 64, 72,84, 136, 144, . . .References[1] S.M. Buckley and D. MacHale: Conjugate Deficiency in Finite Groups, Bul-letin of the Irish Mathematical Society, 71 (2013), 13–19.[2] S.M. Buckley, D. MacHale and A. Nı́ Shé: Degree Sum Deficiency in FiniteGroups, Mathematical Proceedings of the Royal Irish Academy, Vol. 115A No.1 (2015), 1-11.[3] J.D. Dixon: Problems in Group Theory, Dover Publications, 2007.[4] P.X. Gallagher: The Number of Conjugacy Classes in a Finite Group, Math-ematische Zeitschrift, Vol. 118 No. 3 (1970), 175–179.[5] W. Lederman: Introduction to Group Characters, Cambridge University Press,1987.[6] I.D. MacDonald: The Theory of Groups, Clarendon Press, Oxford, 1968.[7] D. MacHale: How Commutative Can a Non-Commutative Group Be?, TheMathematical Gazette, Vol. 58 No. 405 (1974), 199–202.[8] A. Mann: Finite Groups containing Many Involutions, Proceedings of theAmerican Math. Soc., Vol. 122 No. 2, October (1994), 383–385.[9] D.J.S. Robinson: A Course in the Theory of Groups, Springer, 1993.Brian Dolan is a mathematical graduate of University College Cork. He workscurrently in Computer Science in the UK.Des MacHale is Emeritus Professor of Mathematics at University College Corkwhere he taught for nearly forty years. His mathematical interests are in abstractalgebra but he also works in number theory, geometry, combinatorics and the70 DOLAN, MACHALE AND MACHALEhistory of mathematics. His other interests include humour, geology and words.Peter MacHale is the Systems Manager in the Insight Centre for Data Ana-lytics, Computer Science Department in University College Cork. His interestsare constraint programming and graph theory. His other interests include music,science fiction and gaming.(Brian Dolan and Des MacHale) School of Mathematical Sciences, Uni-versity College Cork(Peter MacHale) Insight Centre for Data Analytics, University Col-lege CorkE-mail address : curlyjim@gmail.com, d.machale@ucc.ie, p.machale@ucc.ie

Counting commutativities in finite algebraic systems

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Counting commutativities in finite algebraic systems

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