Groups formed by redefining multiplication

Canad. Math. Bull. Vol. 31 (4), 1988 GROUPS FORMED BY REDEFINING MULTIPLICATION BY K. A. CHANDLER ABSTRACT. Let G be a group with elements 1,. . . , n such that the group operation agrees with ordinary multiplication whenever the ordinary product of two elements lies in G. We show that if n is odd, then G is abelian. 1. Introduction. In the American Mathematical Monthly [1], Rodney Forcade, Jack Lamoreaux, and Andrew Pollington conjecture that it is possible to form a multiplicative group on the set {1, 2 , . . ., n) so that for any two integers whose product under ordinary multiplication is k ^ n, their product in the group is k. Further, they believe, but have not proved, that such a group (which we shall refer to as an FLP group) must be abelian. In this paper we prove the following: THEOREM. Every FLP group of odd order is abelian. 2. For sufficiently large groups, the theorem holds. Define H(n, y) to be the set of positive integers less than or equal to n all of whose prime factors are less than or equal to y. Let ^(n, y) be the number of elements of H(n, y). 1. Let n be odd. If there is a number r such that ^(n, r) > n/3 and n/r) > n/9, then any FLP group G of order n is abelian. LEMMA ^(n, PROOF. Let q be a prime %nlr. Then for any prime p ^ r, qp ^ qr ^ n, hence q • p = p • q where • denotes the group product. Thus q commutes with every prime in H (n, r), and therefore with every element of H(n, r). If Z(q) is the set of elements which commute with q, \Z(q)\ > n/3. Since Z(q) is a subgroup of G, and G has no subgroups of index 2, Z(q) = G. That is, q is in the centre, Z(G), of G. Hence, H(n, r) is contained in Z(G). Now, [G:Z(G)] < 9 and is odd, hence G/Z(G) is cyclic, and so G is abelian. • Direct computation of ^ shows that all FLP groups of odd order ^=23000 are abelian. Received by the editors November 4, 1986, and, in revised form, July 3, 1987. AMS Subject Classification (1980): 20F05, 10H15. © Canadian Mathematical Society 1987. 419 Downloaded from https://www.cambridge.org/core. 01 Mar 2022 at 20:35:12, subject to the Cambridge Core terms of use. K. A. CHANDLER 420 [December More general results on the function ^(n, y) can be obtained by introducing a function p(u) (the Dickman function), which is the continuous solution of the differential difference equation: p(u) = 1, for 0 ^ u â 1 up'(u) = —p(u — 1), for u > 1. It is known that p is strictly decreasing for u ^ 1, and that 0 ^ p(u) ^ 1. In [2] Ramaswami proves that, given c > 0 there is a constant K > 0 such that for all u â c and all « ^ 1, ¥(n, « i/M ) = n(p(u) ± K/\og n). Notice that for 1 ^ u ^ 2, p(w) = 1 - log w, and thus p(e 2/3 ) = 1 — log(e / 3 ) = 1/3. Since p is strictly decreasing for u â 1, for any 1 ^ w < e 2 / 3 , and for n sufficiently large, we have ^ ( H , n]/u) > n/3. Using the trapezoidal rule to estimate the required integral, we have shown that p(e2/3/(e2/3 - 1) ) > .28038 > 1/9, and so we can use Lemma 1 with r = nxiu, where u = e — e, for e > 0 and sufficiently small, together with Ramaswami's formula, to obtain the next lemma. LEMMA 2. For odd n sufficiently large, any FLP group of order n is abelian. 3. Finding the bound. To determine how large is "sufficiently large" in Lemma 2, we must examine the proof of Ramaswami's theorem using approximations for some number-theoretic quantities. In all that follows, p denotes a prime, and c = a ± b means that a — b ^ c ^k a + b. LEMMA 3. (a) If x > 1, TT(X) < 1.2555(x/log x), where TT(X) is the number of primes (b) There is a constant b such that ^x. log log x + b < 2 - < log log x + b + ~, for 1 < x ^ 108, P^xP (logx) z and 2 - = log log x + b- ± 2(logx)~> for 2 -«---' P^xP x = 286 - For proofs of these, see [3]. Note that, in particular, if x > 1 and 1 ^ ux ^ u ^ ux + 1, then l og JL H, u l— ^ (log x ) 2 2 x "»< p s*"»' - ^ log - + "' p ux + U \ 2(log x ) 2 Downloaded from https://www.cambridge.org/core. 01 Mar 2022 at 20:35:12, subject to the Cambridge Core terms of use. 1988] 421 GROUPS LEMMA 4. (a) For u ^ 1, *(JC, xx/u) (b) For 1 ^ u ^ 2, = [x\. *(*, * 1/M ) = [JC] - 2 [*//>] *1/M</>^JC (c) Let w, ^ 2. Suppose there exist positive constants X and K0 such that for all v ^ ul9 *(JC, for all x i? X x1/v) = JC( '. Then for any u such that ux ^ M Si w, + \,andfor all x iË X, *(x, x 1 / u ) = xp(M) + *(x, x l / u ') I PROOF, logxl logxx xp(Ul) 2 ((logx) l o g x ) // \M,/ 2 3(log;c) 3(log x) (a) and (b) are immediate. From [2], we get that for i/u *(x, x ) = xp(u) + *(x, x 1/u i) - M J ^ 2 xp(Ul) It) I ° x"«s,s»"«i log(x//>) I J (log 0 2 " x log ? f fc(0 /iogç^O\l* 11 l l ( l o g ?2P ) 2v \ log/ log? )K"I)' Vgo where k{t) is defined by &(?)/(log t)2 = S^g, \/p — log log t — b. We have JC 1 / M ^/^JC 1 / M ' l o g ( * / / ? ) M = 2 ! 1A0, l o g JC j C 1 / M ^ / ^ * , / M ' /> "l*0; Uog— ± -J , logx\ Wj (logx)/ by Lemma 3(b), while 1/*;;: _*«, l-'' (log/) I-'* 2= log x j« ' log? / ((log?) i o g ? r2 •J*"" \ log? log / log, / A \ log? ?(log? (log ?r , since /log(x/Q\ I log / / 1 for Downloaded from https://www.cambridge.org/core. 01 Mar 2022 at 20:35:12, subject to the Cambridge Core terms of use. 422 [December K. A. CHANDLER t < xuu' log* 1 = 3(io g tyi* r l/K. — «i u 3(log x)2' Finally, by Lemma 3(b), [ k(t) /log(x/OV, j / « . l ( l o g 0 2 ^ log/ (log x)2' nx since if JC1/M> ^ 108 then 0 < k(xVu) < 1 and 0 < k(xl/u') < 1, while if JC17"1 > 108 then xl/u > 104 > 286, so that \k(xl/u) \ < 1/2 and \k(xl/u*) | < 1/2. • Now, for 1 ^ w ^ 2, *(x, xl/u) = [x] - 2 [x/p] xl/u<p^x = [x] — x ZJ xl/u<p^xP - + ^, [x/p] xUu<p^x \p So, by Lemma 3(b), for 1 ^ w ^ 2, *(x, x 1/M ) > jcUii) " + ! 2(log xf ^ x p(w) - x 1 2.5 (log x) 2 while *(x,xuu)A/u < x\p(u) 2 + <*)• (log xY Thus, (1) *(JC, x 1/M ) = x(p(u) ± ^ ^ for x ^ e5, 1 ^ u ^ 2, \ log x / by Lemma 3(a) and (2) *(JC, xl/u) > x(p(u) - .02505), for x ^ e10, 1 ^ w ^ 2. Then in Lemma 4(c), if we take X = e10, K0 = 2.0555, and ux = 2 we have for 2 â u ^ 3 and JC â e10, ^(JC, x 1/w ) = xp(n) + llogxx *(JC, V*) - *p(2) \ux1 (log * y + 3u2 A- u3 - u] 3(log x) 2 Downloaded from https://www.cambridge.org/core. 01 Mar 2022 at 20:35:12, subject to the Cambridge Core terms of use. 1988] 423 GROUPS > xLu) - .02505 - ^Hifiogf") + _Jli_) 2 I logjcl \2> (log*) / - 3 ' 1 + ',7'U(i). Therefore, if 2 ^ w ^ 2.12 (3) ¥(*, x 1/M ) ^ x(p(w) - .1176), for JC ^ e10. Thus, for n ^ ew, in order to get ^(n, nx/u) > n/3, it suffices that p(w) ^ 1/3 + .02505, by (2). Since p(u) = 1 - log u for 1 ^ w ^ 2, we get p(w) ^ 1/3 + .02505 for 1 ^ w ^ exp(l - 1/3 - .02505), in particular for u = 1.897. Now we will use (3) to show that for n ^ e10, ^(«, wO-897-i)/i.897) > n/g by showing that p(1.897/.897) - .1176 > 1/9. Here, 2 < 1.897/.897 < 2.115, so p(1.897/.897) > p(2.115), and by approximating the integral we get that p(2.115) - .1176 > .2540 - .1176 = .1364 > 1/9. Hence, for n > e10, we get that for u0 = 1.897, *(n, nl/u°) > n/3, while ^(w, w(Mo-i)/"o) > n/9m W e have verified the theorem directly for n ^ 23,000 using Lemma 1, so the proof of the theorem is now complete. This research was supported by a Natural Sciences and Engineering Research Council of Canada University Undergraduate Summer Research Award at Dalhousie University, Halifax, N.S. ACKNOWLEDGEMENTS. REFERENCES 1. Richard Guy, éd., A group of two problems in groups, Amer. Math. Monthly 92 (1986), pp. 119-121. 2. V. Ramaswami, On the number of positive integers less than x and free of prime divisors greater than xc, Bull. Amer. Math. Soc. 55 (1949), pp. 1122-1127. 3. J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), pp. 64-92. UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, B.C. V6T 1Y4 Downloaded from https://www.cambridge.org/core. 01 Mar 2022 at 20:35:12, subject to the Cambridge Core terms of use.