Drápal, Aleš

Kepka, Tomáš

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeAleš Drápal; Tomáš KepkaA note on the number of associative triples in quasigroups isotopic to groupsCommentationes Mathematicae Universitatis Carolinae, Vol. 22 (1981), No. 4, 735--743Persistent URL: http://dml.cz/dmlcz/106115Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1981Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 22.4 (1981) A NOTE ON THE NUMBER OF ASSOCIATIVE TRIPLES IN QUASIGROUPS ISOTOPIC TO GROUPS Ales DRAPAL, Tomds KEPKA Abstract: Let G be a f i n i t e non-associative quasigroup of order n l so top ic to a group. Denote by a(G) the number of asdociative t r i p l e s of elements of G. Then a(G),*-.n3 - 4n • • 6n, provided n £ 3 i s odd, and a(G)£n* - 4n2 • 8n, provid-ed n i s even. Key words: Aasociative t r i p l e , quasigroup, isotopy. Class i f i ca t ion: 20H05 In the past , some problems concerning associat ive t r i p -l e s of elements in f i n i t e groupolds were studied from time to time (see [ 1 3 , {.31,1.4-1 and 153). Such questions and inves -t igat ions ( e spec ia l ly those concerning enumerations) belong to a certain branch of combinatorial algebra, and therefore they are of Interest in the present time, too. In C2l, the upper and the lower bounds for a(G), G being a f i n i t e non-as-soc ia t ive commutative quasigroup i so top i c to a group, were found. The purpose of t h i s short note i s to invest igate the same problem in the non-commutative case. 735 1. IntroJUgliiaa- For a groupoid G, let A(G) « 4(a,bfc); afbfc6G, a.bc « ab.c} and a(G) « card A(G)# Let C be a class of groupoids. Then, for every positive integer nf we define numbers a(Cfn) and b(C,n) as follows: a(C,n) « -1 « b(Cfn) if C contains no groupoid of order n; a(Cfn) « min a(G), G«s.C, card G = nf if C contains at least one groupoid of order n; b(C,n) « n^ if C contains at least one groupoid of order n and every such groupoid is associative; b(Cfn) « max a(G)f G e C, G not associative, card G « n, if C contains at least one non-associative groupoid of order n. Let G be a groupoid and aeG. Define two transformations La and R0 of G by L (b) « ab and R (b) « ba. a a a a 2* Arall-Tary results. In this section, let G<+} be a fi-nite group (possibly non-commutative) of order n and f a per-mutation of G. Put f'(x) * f(x) - x and f "(x) » -x • f(x) for every X G G . Let p(f) * card 4(xfy); xfy eGf f'(x) « f'(y)l and q(f) « card 4(x,y); x9yeG, f"(x) * f"(y)K For every a e Im f' Caelm f " ) , let p(aff) « card A (q(a,f) « card A) where A is the block of ker f' (ker f") with f'(A) » a (f"(A) • a). If a 41* t' (a 41m f"') then p(a,f) « 0 (q(aff) « 0). 2.1. Lemma. p(f) * \ p(a,f)2 and q(f) a |L q(aff)2. Prbof. Obvious. 2.2. Lemma. p(f) * p(L*f) and q(f) « q(R*f) for every aeG. Proof. Obvious* 2.3. Lemma. Suppose that n is odd and f4-L^ (f#Rg ) 736 -for every aeG* Then p(f)^n2 - 4n • 6 (q(f)^n2 - 4n • 6). Proof. The proof ia in fact the same as that of £2, Lemma 2.33. 2.4. Lemma* Suppose that t^^B (f4-R*) for every aeG. p(f)£ n2 - 4n • 8 (q(f)£ n2 - 4n • 8). Proof. The same as that of £2, Lemma 2.33. 3. Auxiliary reaulta. In this section, let G(+) be a finite group of order n and f, g permutations of G. Put r(ffg) « eard*(afb); afbeGf f"(a) -*g"(b)Jf B(ffg) » «4(afbfc); -a • f(a) - f(b) *-g(b) • g(c) - cjf a(ffg) « * card B(ffg) and t(f,g) -» card{(a,b); g~X(-f"(a)) « « f'^-g'fb))! - cardf (afb); f(b) • g(b) » bf f"(a) « *f"(b)l - card-f(afb); a^b, f(a) • g(a) * af g'(a) « « g'(b)}. 3.1* Lemma. r(ffg) * 2- p(afg)q(aff). Proof. Obvious. 3.2. Lemma. Let nZl and let alt... ,an,blf... tbn be re-al numbers. Then Z a^b^ max C Z af, Z b 2). Proof. Obvious. 3.3* Lemma, r(f,g)^ max (q(f)fp(g)). Proof. Use 3.1, 3.2 and 2.1. 3.4. Lemma. t(ffg) • p(g) • q(f) - n£s(ffg). Proof. Put A « $ ( a f a f a ) ; aeGi, B «4(a f b,b) ; a # b f f " ( a ) « f " ( b H f C M ( a f a f b ) ; a*b, gH&~iF^mi and D » { ( a f c f b ) ; c » g ^ - f ' ^ a ) ) » f - ^ - g ' f b ) ) ! . Then AuBuCoV£B(ttg)t AAB * Ar»C » B A C * lSf card A * nf - 737 card B » q(f) - n, card C * p(g) - n. Finally, Dn(AuBuG) » * 4(a,b,b); f(b) • g(b) a b, f "(a) » f "(b)>Jut(a,a,b); f(a) • g(a) * a, a4-b, g'(a) -» g'(b)i. 3.5. Lemma. Suppose that neither f" nor g' is a permu-tation. Then n • 4 . £ s ( f , g ) , Proof. q(f)£ n • 2 and p(g)^n • 2. 3.6. Lemma. Suppose that either f" or g is a permuta-tion. Then n _= s(f,g). Proof. Let f" be a permutation. Then, for all b,ccG, there is an aeG with -a • f(a) * -g(b) • g(c) - c • f(b). 3.7. Lemma. s(f,g) - ^ r(ICf(D)f» ^ ( b ) ^ s * « J * ^(a»R-f(b)f)P(a'L-g(b)g)-Proof. Easy. 3.8. Lemma. s(f,g)£ n«max(q(f),p(g)). Proof. By 3.3 and 2.2, r(R-£(D)*» LtgCb)g)- m a x ( q ( f ) , p ( g ) ) and we can use 3.7. 3.9. Lemma. If f « id (g * id) then s(f,g) * np(g) (s(f,g) » nq(f)). Proof. Easy. 4. Auxiliary results. In this section, let G(^) be a f i -nite group of order n and f, g permutations of G. Define a mul-tip l ication on G by ab * f(a) • g(b). In this way, we obtain a quasigroup G. 4 .1 . Lemma, ( i) G contains a left unit i f f g * Lfi for some aeG* 738 -(11) G contains a right unit Iff f » 1* for soma aeO, ( i l l ) G la a group Iff f » Rj and g « i j for aoaa a fbeG. I*oof. Easy. 4.2. Lemma. a(G) » a ( f ,g ) . Proof. (x fy,s)€ A(G) i f f f(x) • g(fCy) • g(e)) » f ( f (x)* • g(y)) • g(z) . Since ff g ara permutationa, a(G) « card A, where A »«C(xfyfs); x • g(f(y) • «) * f (x • g(y)) • *J. -Defi-ne a sapping h of A Into B(f,g) by hCxfyfs) * (x • g(y) , y,f(y) • «)• Then h la bijective. 5. au^glgoUPg iff9twig fa fflr9VPg> In the following re-sult , l e t a(n) ** a(C,n) and b(n) • b(C.n) where C la the d a as of left loops iaotopic to groups. 5.1. Theorem, ( i ) a(l) • 1 « b ( l ) f *(Z) » 8 * b(2). ( i i ) a(n) -* n for every n auch that either n la odd or n is divisible by 4. ( i l l ) a(n) m n • 2n for every even n not divisible by 4. (iv) b(n) « tr - 4n • 6n for every odd n > 3 . "\ 9 (v) b(n) « tr - 4n + 8n for every even n. Proof. (1) These equalities are clear. (11) and (ill). Let G be a finite quaaigroup of order n auch that G contains a left unit e and G is isotopic to a group. Put x • y « R~ (x)y for all xfy€G. Then G(+) ia a group and xy « f(x) • y, f a R0. By 4.2, and 3.9, a(G) -* nq(f). Since 9 9 n£q(f), n . s a(G) and we have proved that n .6 a(n). If n * 2m for an odd m then f" cannot be a permutation (this fact ia eaay and well known - see e.g.EU), and ao n • J>£q(f)f 739 -n * 2n^a(G) and n • 2n:-=a(n). In the r e s t , we can proceed s imi lar ly as in 12 , Lemmas 1 .5 , 1.6, 2 .103 . ( i v ) and ( v ) . Suppose that n > 3 . Let G be a non-associ-ative f i n i t e quasigroup of order n such that G contains a l e f t unit and G i s i s o t o p i c to a group. Then there are a group G(+) and a permutation f of G such that xy * f (x ) • y for a l l x , y c G . Since G i s not a s s o c i a t i v e , f4=Ra for every » £ G . By 4 . 2 , 3 .9 and 2.4 (reap . 2 . 3 ) , a (G)^n 3 - 4n 2 • 8n (reap . a ( G ) £ n J - 4n + 6n provided n i s odd). In the r e s t , we can proceed s imi lar ly as in C2, Lemmas 2 .6 , 2 . 7 3 . In the fol lowing r e s u l t , l e t a(n) =- a(C,n) and b(n) * * b(C,n) where C i s the c lass of quasigroups i s o t o p i c to groups. 5 . 2 . Theorem, ( i ) a d ) » 1 -* b ( l ) , a(2) » 8 » b ( 2 ) . ( i i ) n •»• 4 £ a ( n ) - = n for every n > 2 such that n i s e i ther odd or d i v i s i b l e by 4 . Ci i i ) n + 4 £ a(n)^= n + 2n for every even n which i s not divisible by 4. 3 2 ( iv) b(n) a n - 4n • 6n for every odd n£"3. ** *% (v) b(n) s n - 4n • 8n for every even n. Proof, ( i i These e q u a l i t i e s are c lear . ( i i ) and ( i i i ) . We can assume that 3 ^ n. Then n • 4 6 n and the resu l t fol lows from 3 . 5 , 4.2 and 5 . 1 . ( iv ) Let G be a non-associative quasigroup of order n such that G i s I sotop i c to a group. With respect to 5 . 1 , we can assume that G i s neither a l e f t nor a right loop . Then the -re are a group G(+) and permutations f, g of G such that ab » » f (a ) • g(b) and f4=Rt » g ^ t * ° r a 1 1 a ,b€G (use 4 . 1 ) . By 8 8 4,2, 3.8 and 2.3, a(G)£ n3 - 4n2 • 6n. Thus a(n)^n3 - 4n2+ 6n. 740 -The converse inequality follows from 5.1(iv). (v) We can proceed similarly as in (iv). 6. Auxiliary results. In this section, let G(+) be a fi-nite abelian group of order n and f, g endomorphisms of G(+). Put C(f,g) *4(a,b); a,beGf f(a) * g(b)$ and u(ffg) « » card C(f,g). 6.1. Lemma. n^u(f,g). Proof. Define a mapping h:G*<G —> G by h(a,b) » f (a ) -- g (b) . Then h i s a homomorphism of the abelian group G(+) x x G(+) into G(+) and Ker h » C( f ,g ) . Hence (card Ker h)» .(card Im h) « n and card Im h £ n . Consequently, n £ u ( f f g ) . 6 . 2 . Lemma. Suppose that n » 2m where m21 i s odd and that f » h ' f g * k ' for some automorphisms h and k of G(+). Then 2 n £ u ( f f g ) . Proof. We can assume that G(+) « H(+ix K(+), h » h^x h^f k » k-̂ x kg where H(+) i s a group of order m, K(+) » - l O f l } i s a two-element group , h^fk^ are automorphisms of H(+) and h.̂ , kg of K(+). Then h2 =* id « k? f h^ » 0 » k^f f * h ^ O , g * « k£x 0 and the resu l t fo l lows e a s i l y from 6 . 1 . Put r » card Ker f and s » card Ker g. 6.3. Lemma. u(ffg)*£max (rn,sn). Proof. For every acG, let r(a) * card4b; f(b) » »} and s(a) » card -f b; g(b) » a}. Then u(f,g) » -^ r(a)a(a)» By 3.2, u ( f f g ) ^ m a x ( . 2 : r(a)2>f -£ s(a)2). However, m « Z r ( a ) 2 and sn »-£" a(a) • 6.4. Lemma. Let f * 0 (g ** 0). Then u(f fg) « an - 741 (u(f,g) * rn). Proof. Easy. 7* Auxiliary reaulta. In this section, let G(+) be a fi-nite abelian group of order n, f, g commuting automorphisms of G(+) and weG. Put ab -» f(a) • g(b) • w for all a,b£G. We obtain thus a medial quasi group G. 7.1. Lemma. (1) f * id iff G contains a right unit, (ii) g • id iff G containa a left unit. (iii) G is a group iff f * id « g. Proof. Ea8y. 7.2. Lemma. a(G) * nu( f ' ,g ' ) . Proof. (x,y,z)c A(G) i f f f^x ) • g(z) • f(w) « f(x) • • g Cz) • g(w). Thus a(G) a n.card A where A » i (x,y); f(x • w) - x - w * g(y + w) - y - w}. The rest i s clear. 8. Medial ouasigroups. In the following result, let a(n) s a(C,n) and b(n) = b(C,n) where C la the class of medial qua-si groups. &•--• Theorem, ( i ) a(n) » n for every n such that n la either odd or divisible by 4. (11) a(n) * 2n for every even n not divisible by 4. ( i i i ) b(l) * 1 and b(n) » n3/p for every odd n2 3, p being the least prime dividing n. (iv) b(2) -* 8 and b(n) • nVp for every n * 2» where m£3 is odd and p la the least prime dividing m. (v) b(n) m n3/2 for every nZ4 divisible by 4. 742 -Proof. Using 6 . 1 , 6 . 2 , 6 . 3 , 6 .4 , 7 . 1 and 7 . 2 , we can proceed s i m i l a r l y as i n the proofs of 5 . 1 , 5*2 and L2, Theo­rem 3 . 3 3 . R e f e r e n c e s [13 A.C. CLIMESCU: Etudes sur l a t h ^ o r i e des systemes m u l t i -p l i c a t i f s uniformes I . L ' i n d i c e de non asso­c i a t i v i t y , B u l l , de l ' E c o l e Polytech. de J e s ­sy 2(1947), 97-121. [ 21 T. KEPKA: Notes on a s s o c i a t i v e t r i p l e s of elements in commutative groupoids ( t o a p p e a r ) . [31 D.A. NORTON: A note on a s s o c i a t i v i t y , P a c i f i c J . Math. 10 (1960), 591-596. [ 43 G. SZASZ: Die Unabhangigkeit der Assoz ia t iv i ta t sbed ingun-gen, Acta S c i . Math. Szeged 15(1953), 20-28. [ 5J A. WAGNER: On the a s s o c i a t i v e law of groups, Rend. Math. e Appl. 21(1962), 60-76. Výzkumný ústav matematických s t r o j ů , Luzná 2, 16000 Praha 6 Československo Matematicko-fyzikální f a k u l t a , Univers i ta Karlova, Sokolovská 8 3 , 18600 Praha 8, Československo (Oblátům 2 9 . 1 . 1981.1 743 -

A note on the number of associative triples in quasigroups isotopic to groups

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A note on the number of associative triples in quasigroups isotopic to groups

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Institute of Mathematics AS CR, v. v. i.