QUASIGROUP AUTOMORPHISMS AND THE NORTON-STEIN COMPLEX

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volum 138, Numr 9, Sptmr 2010, Pgs 3079 3088 S 0002-9939(10)10473-0 Articl lctroniclly pulishd on April 29, 2010 QUASIGROUP AUTOMORPHISMS AND THE NORTON-STEIN COMPLEX BRENT L. KERBY AND JONATHAN D. H. SMITH (Communictd y Jonthn I. Hll) Astrct. Suppos tht d>1 is th lrgst powr of two tht divids th ordr of finit qusigroup Q. It thn follows tht ch utomorphism of Q must contin cycl of lngth not divisil y d in its disjoint cycl dcomposition. Th proof is otind y considring th ction inducd y th utomorphism on crtin orintl surfc originlly dscrid in mor rstrictd contxt y Norton nd Stin. 1. Introduction For n utomorphism of finit qusigroup (or Ltin squr) Q, th multist of cycl lngths in th disjoint cycl dcomposition of th utomorphism is clld th cycl structur of th utomorphism. It is thn nturl to pos th following qustion: Givn positiv intgr n, wht r th possil cycl structurs of utomorphisms of qusigroups of ordr n? Th qustion ws ddrssd in [2] for smll vlus of n nd in [1] for th spcil cs of utomorphisms hving only on nontrivil cycl. Th gnrl cs rmins unsolvd, ut in this ppr w us mthods from lgric topology to rsolv som of th mor difficult css. Algric topology normlly procds y functoril ssocition of lgric ojcts to topologicl spcs. W rvrs this procdur in 2 y constructing n orintl surfc S Q for ch finit qusigroup Q, such tht th ssocition Q S Q is functoril (Thorm 2.3). In 3 w ndow th surfc with th structur of CW-complx. This clrifis nd xtnds construction of Norton nd Stin [6] tht ws limitd to th cs of idmpotnt qusigroups. Using our construction, w prsnt somwht simplifid proof of rsult of Stin [7, Th. 1.1]: Corollry 3.4. Lt Q finit qusigroup, with Q 2 mod 4. Thn th utomorphism group of Q dos not ct trnsitivly. In 4, w stt nd prov our min rsult, which dpnds on lmm stlishd in 5. Th proof of th lmm uss monodromy tchniqu dting ck to Nilsn [5]. Th min rsult is s follows: Rcivd y th ditors Novmr 11, 2009. 2010 Mthmtics Sujct Clssifiction. Primry 20N05, 05B15. Ky words nd phrss. Ltin squr, qusigroup, surfc, monodromy. 3079 c 2010 Amricn Mthmticl Socity This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

3080 B.L. KERBY AND J.D.H. SMITH Thorm 4.2. If 2 l (with l>0) is th lrgst powr of two dividing th ordr of finit qusigroup Q, thn th disjoint cycl dcomposition of ch utomorphism of Q must contin cycl whos lngth is not divisil y 2 l. Som indiction of th scop of this rsult is otind y considring Flcón s dscription [2] of th possil cycl structurs of utotopis of qusigroups Q with Q 11. In tht work, thr potntil cycl structurs of utomorphisms could only ruld out y n xhustiv computr srch. Ths wr 2 3 (i.., cycl structur consisting of thr disjoint trnspositions) for qusigroups of ordr 6, nd 6 1 4 1 nd 2 5 for qusigroups of ordr 10. All ths css r now foriddn y Thorm 4.2 (with l 1). 2. Construction of th Norton-Stin surfc Lt Q qusigroup of ordr n. ThnQ is dtrmind y its (trnry) multipliction tl T Q {(,, c) Q 3 c}. For ch tripl t (,, c) int Q, construct n quiltrl tringl whos vrtics will lld,, ndc, circumnvigting th tringl in th countr-clockwis dirction, nd whos dgs will lld (, ) 0,(, c) 1,(c, ) 2,sshowninFigur 1. c (c, ) 2 (, c) 1 (, ) 0 Figur 1. A sic tringl. Th disjoint union of ths n 2 tringls forms topologicl spc, suspc of th Euclidn pln, which w will dnot y P Q. Not tht y dfinition, ch tringl contins thr distinct dgs, vn though th vrtx lls,, ndc my not ll distinct. An dg with ll of th form (, ) i will clld dgnrt dg, whil th tringl corrsponding to tripl of th form (,, ) will clld dgnrt tringl. Exmpl 2.1. Considr th cyclic group Q of ordr 3. Writ its usul multipliction tl s Q This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

QUASIGROUP AUTOMORPHISMS 3081 Th corrsponding spc P Q is shown in Figur 2. Figur 2. Th spc P Q for th cyclic group Q of ordr 3. For convninc, w hv omittd th full dg lls (, ) i, instd mrking ch dg (, ) i with 0, 1, or 2 slshs ccording s i 0, 1, or 2, rspctivly. If n>0 thr will mny vrtics of P Q with th sm ll. Howvr, th qusigroup condition nsurs tht for ch, Q nd i {0, 1, 2} thr is uniqu dg with th ll (, ) i. Likwis, thr is uniqu dg with th ll (, ) i. Thus for ch,, ndi w my dfin function (th so-clld psting mp) which mps th dg (, ) i isomtriclly onto th dg (, ) i, th img procding clockwis s w procd countr-clockwis in th domin, so tht th vrtx lld is mppd onto th othr vrtx lld, nd th vrtx lld is mppd onto th othr vrtx lld. If w idntify points which corrspond undr th psting mp, thn w otin topologicl quotint spc of P Q which w cll th Norton-Stin surfc of Q nd dnot y S Q. W will omit th proof tht S Q is (usully disconnctd) surfc, sinc this fct is wll known for spcs constructd y psting polygons togthr in such mnnr (s,.g., [4]). Th only issu hr is tht, in th cs of dgnrt dg (, ) i, th dg is pstd onto itslf with opposit orinttion. Topologiclly, this simply collpss tringl into digon formd y th two rmining dgs (s Figur 3). A dgnrt tringl (,, ) collpss furthr from digon into sphr. (S Figur 4, whr th procss is rokn down into two stps.) Th rsulting quotint spc will still surfc. Sinc th dgs com in pirs which hv opposit orinttion undr th psting mps, it follows tht th Norton-Stin surfc is orintl (gin s,.g., [4] for dtils). Exmpl 2.2. For Q th cyclic group of ordr 3, Figur 5 shows th spc rsulting from P Q ftr pplying nough psting mps so tht th spc flls into 3 closd This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

3082 B.L. KERBY AND J.D.H. SMITH c c Figur 3. Collps to digon. Figur 4. Collps from digon to topologicl sphr. polygons, ch dg of which is pird which nothr dg (possily itslf) on th smpolygon.fromthisitisvidntthts Q consists of 3 disconnctd sphrs. Figur 5. Th spc S Q for th cyclic group Q of ordr 3. Th functoril ntur of this construction of th Norton-Stin surfc my dscrid s follows: Thorm 2.3. A homomorphism ψ : Q 1 Q 2 of qusigroups inducs continuous mp ψ : S Q1 S Q2 on th corrsponding Norton-Stin surfcs. This corrspondnc dtrmins functor from th ctgory of finit qusigroups nd homomorphisms to th ctgory of surfcs nd continuous mps. Proof. First, ψ dtrmins mp ψ : P Q1 P Q2 y snding ch tringl (,, c) of P Q1 isomtriclly onto th corrsponding tringl (ψ(),ψ(),ψ(c)) of P Q2,such This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

QUASIGROUP AUTOMORPHISMS 3083 tht th vrtics lld,, ndc mp to th vrtics lld ψ(), ψ(), nd ψ(c), rspctivly. Thn ψ mps ch dg (, ) i isomtriclly onto th dg (ψ(),ψ()) i. It follows tht if f 1 is th psting mp twn two dgs (, ) i nd (, ) i of P Q1,ndx, y P Q1 r such tht f 1 (x) f 1 (y), thn w lso hv f 2 (ψ(x)) f 2 (ψ(y)), whr f 2 is th psting mp twn th dgs (ψ(),ψ()) i nd (ψ(),ψ()) i of P Q2.Thusψinducs wll-dfind continuous mp ψ from th quotint spc S Q1 to th quotint spc S Q2. Tht this corrspondnc dtrmins functor is strightforwrd to vrify. Corollry 2.4. For n utomorphism ψ of finit qusigroup Q, th ordrs of ψ nd ψ r qul. Proof. Suppos tht ψ nd ψ hv rspctiv ordrs n nd m. Thnsincψ n 1, Thorm 2.3 implis ψ n 1. Hnc m n, nd in prticulr m<. On th othr hnd, for ch lmnt q of Q, choosvrtxx q in S Q with ll q. Now ψ m (x q )x q,soψ m (q) q. Sinc th qulity ψ m (q) q holds for ch lmnt q of Q, whvψ m 1ndthusn m. 3. Th Norton-Stin complx nd som pplictions W now giv S Q th structur of CW-complx. Th 0-clls r dfind to th imgs of th vrtics of P Q. Th 1-clls r th imgs of th nondgnrt dgs, nd 2-cll is givn y th img of th intrior of tringl of P Q togthr with ny dgnrt dgs of th tringl. With this ddd structur, w cll S Q th Norton-Stin complx of Q. W ssign th sm lls to th 0-clls of S Q s th corrsponding vrtics of P Q. This is wll-dfind sinc only vrtics with th sm ll r idntifid y th psting mps. Givn nondgnrt dg (, ) i of P Q, w ssign th ll {, } i to th corrsponding 1-cll in S Q. Not tht if n>0, S Q will not simplicil complx, sinc it is possil tht two distinct 2-clls my incidnt with th sm thr 0-clls, nd som 2-clls will incidnt with only two 0-clls nd 1-clls (if th corrsponding tringl contins dgnrt dg). In th cs of dgnrt tringl (,, ), th corrsponding 2-cll is incidnt with only on 0-cll nd no 1-clls. Although only vrtics of P Q with th sm lls r idntifid in S Q,itis notncssrilythcsthtll vrtics with th sm lls r idntifid. Th numr of distinct vrtics of S Q with th ll q is dnotd y Φ(q). It is clr tht Φ is invrint undr isomorphisms. Thus if ψ : Q 1 Q 2 is n isomorphism of qusigroups, thn Φ(ψ(q)) Φ(q). Th numr Z Q q Q Φ(q) is thn n invrint of th qusigroup Q. WsthtZ Q is simply th numr of vrtics in th Norton-Stin complx of Q. Th following rsult is thn immdit. Thorm 3.1. For finit qusigroup Q, th Eulr chrctristic of S Q is χ Z Q 3n(n 1)/2+n 2. This lds us to th following thorm, provd y Norton nd Stin [6] in th spcil cs of idmpotnt qusigroups. This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

3084 B.L. KERBY AND J.D.H. SMITH Thorm 3.2. For qusigroup Q of ordr n, (3.1) Z Q n(n +1)/2 mod 2. Proof. W hv Z Q 3n(n 1)/2+n 2 n(n 1)/2+n n(n +1)/2 mod 2 y Thorm 3.1, sinc th Eulr chrctristic of th orintl surfc S Q is vn. Rmrk 3.3. Th formul (3.1) diffrs from tht of [6, Th. II], cus our dfinition of Z Q includs th vrtics which r ssocitd with th dgnrt tringls (,, ). As shown y th following corollry, vn th smingly miniscul informtion out th Norton-Stin complx providd y Thorm 3.2 cn ld to powrful conclusions out th ssocitd qusigroup. Corollry 3.4 ([7]). Lt Q finit qusigroup, with Q 2 mod 4. Thn th utomorphism group of Q dos not ct trnsitivly. Proof. Suppos tht Q hd trnsitiv utomorphism group. Thn thr would constnt c such tht Φ(q) c for ll q Q. Hnc Z Q q Q Φ(q) c Q 0 mod 2. Howvr, if Q 4k + 2 for som nturl numr k, Thorm 3.2 implis Z Q (2k + 1)(4k +3) 1 mod 2. Rmrk 3.5. () Stin s originl proof of Corollry 3.4 rquird n initil rduction to th cs of idmpotnt qusigroups. () Stin [8, Th. 9.9] osrvd tht for ch positiv intgr n not congrunt to 2 modulo 4, thr is distriutiv qusigroup of ordr n with trnsitiv utomorphism group (compr Fit s rviw of [3]): If n fctors into powrs of distinct prims p 1,p 2,...,p k s p 1 qusigroup k j1 1 p 2 2...p k k, n xmpl is givn y th dirct product ( GF(p j j ), j) with x j y α j x +(1 α j )y, whrα j is con- ) {0, 1}. stnt ritrrily chosn from GF(p j j Th sm tchniqu s in th proof of Corollry 3.4 srvs to giv th following spcil cs of our min rsult. Corollry 3.6. Lt ψ n utomorphism of qusigroup Q, whr Q 2 mod 4. Thn th disjoint cycl dcomposition of ψ contins cycl of odd lngth. Proof. If ll th cycls of ψ hv vn lngth, thn th lmnts of Q fll into orits of vn lngth undr ψ. Th function Φ is constnt on ch orit. This implis tht Z Q Φ(q) 0 mod 2, q Q gin contrry to th impliction Z Q 1 mod 2 of Thorm 3.2. As mntiond in th introduction, thr r thr impossil cycl structurs of utomorphisms which hd to ruld out y xhustiv computr srch in [2]: 2 3, 6 1 4 1,nd2 5. All thr of ths css r now foriddn y Corollry 3.6. Howvr, th mor lmntry tchniqus of this sction r not cpl of liminting such cycl structurs s 4 3 (for qusigroups of ordr 12) or 4 5 (for qusigroups of ordr 20). Furthr progrss rquirs th full gnrlity of our min rsult, dscrid in This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

QUASIGROUP AUTOMORPHISMS 3085 th nxt sction. This ntils considrly dpr ppliction of th Norton-Stin complx. 4. Th min rsult Givn n utomorphism ψ of finit qusigroup Q, Thorm 2.3 yilds continuous utomorphism ψ of th Norton-Stin surfc S Q. Th group ψ gnrtd y ψ thn cts y prmuting th connctd componnts of th surfc S Q. W my thus stt th following lmm. Lmm 4.1. Lt ψ n utomorphism of finit qusigroup Q, nd suppos tht th disjoint cycl dcomposition of ψ consists ntirly of cycls of lngth 2. Suppos tht connctd componnt W of th Norton-Stin surfc S Q is fixd stwis y ψ. Th Eulr chrctristic of W is thn divisil y 2. Th proof of Lmm 4.1, givn in th nxt sction, is t th cor of our min rsult. Hr, w stt th min rsult nd show how it follows from Lmm 4.1. Thorm 4.2. If 2 l (with l>0) is th lrgst powr of two dividing th ordr of finit qusigroup Q, thn th disjoint cycl dcomposition of ch utomorphism of Q must contin cycl whos lngth is not divisil y 2 l. Proof. Suppos tht ll th cycls of n utomorphism ψ hv lngth divisil y 2 l. First, w clim tht it suffics to rstrict ttntion to th cs whr ψ consists of cycls ll of lngth prcisly 2 l. Lt c 1,...,c r th cycl lngths of ψ, nd st c lcm(c 1,...,c r ). Writ c 2 k with k odd. By rplcing ψ with its powr ψ k, w my ssum tht ch cycl of ψ hs lngth powr of two, ch cycl hving lngth t lst 2 l. Sinc 2 l is th lrgst powr of two dividing Q nd th cycl lngths dd to Q, it follows tht ψ must hv n odd numr of cycls of lngth 2 l. Thstoffixdpointsof ψ 2l forms suqusigroup H of Q, onwhichψ cts s n utomorphism consisting of n odd numr of cycls ll of lngth prcisly 2 l,sotht2 l is th lrgst powr of two dividing th ordr of H. Rplcing Q y H nd ψ y ψ H, this provs th clim. Sinc th orits of ψ prtition Q into orits of lngth 2 l, it follows tht Z Q q Q Φ(q) is divisil y 2l. Hnc y Thorm 3.1 th lrgst powr of 2 dividing th Eulr chrctristic of S Q is 2 l1. W will otin contrdiction y showing tht th Eulr chrctristic of S Q is ctully divisil y 2 l. Aswhvsn, ψ cts y prmuting th connctd componnts of S Q.LtW such componnt of S Q. Its stilizr in ψ is gnrtd y ψ 2 for som l. So th orit of W undr ψ consists of 2 componnts, ch with idnticl Eulr chrctristic. Sinc ψ 2 is n utomorphism of Q consisting of cycls of lngth 2 l, Lmm 4.1 implis tht W hs Eulr chrctristic divisil y 2 l. It follows tht th union of th componnts in th orit of W hs Eulr chrctristic divisil y 2 l. Hnc S Q itslf hs Eulr chrctristic divisil y 2 l, th dsird contrdiction. 5. Proof of th lmm In th proof of Lmm 4.1, w spciliz tchniqus usd y Nilsn in [5, 1 4]. Lt ψ n utomorphism of finit qusigroup Q, nd suppos tht th disjoint cycl dcomposition of ψ consists ntirly of cycls of lngth 2. Furthr, suppos tht W is connctd componnt of S Q which is invrint undr ψ.st This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

3086 B.L. KERBY AND J.D.H. SMITH G ψ. Not tht G cts on W y prmuting its points. By Corollry 2.4, w hv G ψ ψ 2,sothtchG-orit on W is finit, of lngth dividing 2. Mor prcis informtion on th orit lngths is givn y th following: Lmm 5.1. Considr th ction of G on th surfc W. () Ech intrior point of 2-cll lis in G-orit of lngth 2. () Ech 0-cll lis in G-orit of lngth 2. (c) Ech point x on 1-cll lis in G-orit of siz 2,with (5.1) 1. Th lowr ound 1 in (5.1) cn only ttind if x is th midpoint of 1-cll. Proof. (): Suppos tht som powr ψ k fixs n intrior point x of som 2-cll (u, v, w). Sinc ψ mps 2-clls to 2-clls, ψ k must fix th 2-cll (u, v, w) stwis. By th construction of ψ k from Thorm 2.3, it follows tht ( ψ k (u),ψ k (v),ψ k (w) ) (u, v, w). In prticulr ψ k (u) u, which implis 2 k. (): If ψ k mps 0-cll u to itslf, thn in prticulr ψ k mps x to vrtx with th sm ll s x. Ifx hs ll q, thn w must hv ψ k (q) q, ndso2 k. (c): Suppos x is point on 1-cll {u, v} i,ndthtψ (x) k x. Thnsincψ mps 1-clls to 1-clls, ψ k must mp th dg {u, v} i to itslf: {ψ k (u),ψ k (v)} i {u, v} i. Hnc ψ k ithr fixs u nd v, or swps thm. In ithr cs, w hv ψ 2k (u) u, so2 1 k. If th fixd point x of ψ k is not th midpoint of th dg {u, v} i,thnψ k cnnot swp u nd v. Thusψ k (u) u nd 2 k in this cs. Dfinition 5.2. Whn th midpoint x of 1-cll lis on G-orit of miniml lngth 2 1, th point x is dscrid s rnch point. Undr th ction of G, th rnch points fll into orits of siz 2 1. Lt s th (possily zro) numr of such orits, so tht thr r 2 1 s rnch points ltogthr. Choos rprsnttiv rnch points x 1,...,x s,onfromch orit. Now, slct disjoint opn discs U 1,...,U s contining x 1,...,x s rspctivly, choosing th U i sufficintly smll so tht th 2 1 s imgs ψ (U k i ) r ll disjoint for 0 k<2 1 nd 1 i s. W rmov ll of ths 2 1 s discs from W nd cll th rsulting spc W 0. Thus W 0 is n orintl surfc with oundry, hving 2 1 s oundry curvs. Lt M th quotint spc W/G of W dtrmind y idntifying points which li in th sm G-orit. Lt π : W M th quotint mp. Th imgs of th discs U 1,...,U s undr π r rmovd from M, ndwcll th rsulting spc M 0. This spc M 0 is lso n orintl surfc with oundry hving s oundry curvs. Th spc W 0 is 2 -fold covring spc of M 0, th rstriction of π to W 0 ing covringmp. Iff :[0, 1] M 0 is ny closd loop in M 0 sd t som point m of M 0,nduis point of W 0 such tht π(u) m, thnf lifts to uniqu pth f :[0, 1] W 0 in W 0 with f(0) u. Sinc th ndpoint of this pth stisfis π( f(1)) m, it follows tht f(1) ψ μ(f,u) (u) for som intgr μ(f,u) modulo 2. Lmm 5.3. Thr is uniqu intgr μ(f) modulo 2 such tht μ(f) μ(f,v) for ll v π 1 {m}. This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

QUASIGROUP AUTOMORPHISMS 3087 Proof. Suppos v π 1 {m}, syv ψ (u). l Sinc ψ l f is th lift of f topth in W 0 strting from v, whv ( ψ μ(f,v) (v) ψ l f ) ( ) (1) ψ l f(1) ( ) ψ l ψ μ(f,u) (u) ψ μ(f,u) ( ψ l (u) ) ψ μ(f,u) (v), whnc μ(f,v) μ(f,u) y Lmm 5.1(). Dfinition 5.4. Th intgr μ(f) modulo 2 of Lmm 5.3 is clld th monodromy numr of th loop f in M 0. Th monodromy numr μ(f) dpnds only on th homotopy clss of th loop f. Furthrmor, for th conctntion f g of two loops f,g sd t m in M, w hv μ(f g) μ(f)+μ(g). Hncwmyconsidrμ s homomorphism from th fundmntl group π 1 (M 0,m)ofM 0 into Z/2 Z. Considr point m on on of th s oundry curvs of M 0.Ltγ loop strting from m tht follows onc round th oundry curv in th positiv dirction. Thr r 2 points of W 0 which mp to m. On th othr hnd, π 1 (γ([0, 1])) is th disjoint union of 2 1 oundry curvs of W 0, which r prmutd trnsitivly y G. Thus ch such oundry curv must contin prcisly two points mpping to m, nd ths two points r intrchngd y ψ 21. This implis μ(γ) 2 1. Now fix n ritrry point m of M 0.Ltγ 1,...,γ s dnot loops tht rspctivly follow onc round ch of th s oundry curvs of M 0 in th positiv dirction. Lt δ 1,...,δ s pths from m to γ 1 (0),...,γ s (0), rspctivly. St γ i δ1 i γ i δ i for 1 i s. Thn μ(γ i )μ(γ) 21 for 1 i s, sincthloopsγ i nd γ i r homotopic. Lt g th gnus of M 0. Thn th fundmntl group π 1 (M 0,m) is gnrtd y crtin loops α 1,β 1,...,α g,β g, togthr with th loops γ 1,...,γ s, sujct only to th rltion (5.2) t 1 t 2 t g γ 1γ 2 γ s 1 with commuttors t i α i β i α 1 i β 1 i. If w pply μ to (5.2), w otin g s μ(t i )+ μ(γ j ) 0 mod 2. i1 j1 Now μ(t i )μ(α i )+μ(β i ) μ(α i ) μ(β i )0for1 i g, ndμ(γ i )21 for 1 j s. Thus 2 1 s 0 mod 2, nd so s is vn. Th Eulr chrctristics of W, W 0,ndM 0 r rltd s follows: χ(w ) 2 1 s χ(w 0 )2 χ(m 0 ). Sinc s is vn, this implis tht 2 χ(w ), s dsird. Acknowldgmnts Th first uthor would lik to thnk Brin Rushton nd Stphn P. Humphris for usful convrstions on th topic of this ppr. This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.

3088 B.L. KERBY AND J.D.H. SMITH Rfrncs [1] D. Brynt, M. Buchnn nd I.M. Wnlss, Th spctrum for qusigroups with cyclic utomorphisms nd dditionl symmtris, Discrt Mth. 309 (2009), 821 833. MR2502191 (2010:20122) [2] R.M. Flcón, Cycl structurs of utotopisms of th Ltin squrs of ordr up to 11, to ppr in Ars Comintori. rxiv:0709.2973v2 [mth.co], 2009. [3] B. Fischr, Distriutiv Qusigruppn ndlichr Ordnung, Mth. Z. 83 (1964), 267 303. MR0160845 (28:4055) [4] J.R. Munkrs, Topology, Prntic-Hll, Englwood Cliffs, NJ, 2000. [5] J. Nilsn, Di Struktur priodischr Trnsformtionn von Flächn, Kgl. Dnsk Vidnskrns Slsk., Mth.-fys. Mddllsr 15 (1937), 1 77, trnsltd s Th structur of priodic surfc trnsformtions, pp. 65 102 in Jko Nilsn: Collctd Mthmticl Pprs,Vol.2, Birkhäusr, Boston, MA, 1986. MR0865336 (88:01070) [6] D.A. Norton nd S.K. Stin, An intgr ssocitd with Ltin squrs, Proc. Amr. Mth. Soc. 7 (1956), 331 334. MR0077486 (17:1043f) [7] S.K. Stin, Homognous qusigroups, Pc. J. Mth. 14 (1964), 1091 1102. MR0170972 (30:1206) [8] S.K. Stin, On th foundtions of qusigroups, Trns.Amr.Mth.Soc.85 (1957), 228 256. MR0094404 (20:922) Dprtmnt of Mthmtics, Brighm Young Univrsity, Provo, Uth 84602 E-mil ddrss: kry@mth.yu.du Currnt ddrss: Dprtmnt of Mthmtics, Univrsity of Uth, Slt Lk City, Uth 84112 E-mil ddrss: kry@mth.uth.du Dprtmnt of Mthmtics, Iow Stt Univrsity, Ams, Iow 50011 E-mil ddrss: jdhsmith@istt.du URL: http://www.orion.mth.istt.du/jdhsmith/ This is fr offprint providd to th uthor y th pulishr. Copyright rstrictions my pply.