Havel, Václav

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

Commentationes Mathematicae Universitatis CarolinaeVáclav HavelOne generalization of planar partitions of Abelian groupsCommentationes Mathematicae Universitatis Carolinae, Vol. 7 (1966), No. 4, 419--427Persistent URL: http://dml.cz/dmlcz/105075Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1966Institute 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.czCommentationea Mathematicae Universitatia Caroliaae 7,4 (1966) ONE GENERALIZATION OP PLANAR PARTITIONS OP ABELIAN V. GROUTS Vaclav HAV.BL, Brno In the present paper, partitions of Abelian groups are investigated which correspond naturally to certain Andre* parallel systems closely related to translation pla-nes* We shall endow these partitions with coordinatizing systems having two compositions which a re shown to be qua-s i f ie lds (the case of translation planes i s characterized by the well-known "planarity" condition)* Definition 1* A gartition P of a nonzero ^ Abe-lian group ( S , 4-) i s a set of nonzero subgroups (S0 +) 7 L e 2 in ( S , +) (called oomponents) where i ) <wcd 7 > / | , ii) S ^ S^ - -f 0 \ for t, * L2 , i l l ) U S « S • t e a "•' The pair i^^S^} i8 said to be generating i f ^ 4 - S * * S . If there exist pairwise distinct indices <A,1 / J ; y e J such that a l l iS^ i ^X * > ^ e ^M°c l *n<* also { S^ 7 ST J are generating pairs, then IP wil l be called a f ^jggEJir tioru If each pair { S^ , SL ] with i^ #* ^ i s generating, (1) A group ( $ ? + ) I 8 s a i (* t 0 ^ IJS25S52 ** <̂ **<a£ S > 1. .The neutral element of an additive group wi l l be denoted by Q0 - 419 -« (2) then P i s called a glanarjgartition Proposition 1 (Andrd). Let IP-={(St,+)l t e J ? be a par-t i t i o n of a nonzero Abelian group ( S , +- ) . Then --J i i (S0+X \ X € S$ \ u € J$ i s a decomposition ( 3* of the set { SL + x I i e J, x € S \ ( 4 ) , i i ) ocutd, (SL + X ) > 1 for a l l I 6 J , X e S , i i i ) there are three elements of S which do not belong to the same SL + X, iv) i f t ^ ; :x —> X + cu i s a ( le f t ) translation of ( S, + ) , then To, ( SL + x )-=r S0 + (x+(l) holds for a l l i,ey, a, e S7 x e S and {%^\ CL € S } acts simply transitively on S P v) i f {S^ 0 S~ \ i s a generating pair, then wed, (CS^ +x )r% CS^ + 'y,))*-1 for a l l X , ty e $ ' Proof: [ 1 ] , pp.163-165 and [ 2J, pp.156-158. r̂ ;f j ^ t j o n g. A o^as^fieid i s a triplet ( F7+r) ;where i ) ( F, +* ) is a nonzero Abelian group ^ , (2) Cf.[3] ,p .75. In Ll3 ,p. l63 , a planar partition i s called a congrue^ge.. (3) A decomposition of a set A 4* 0 i s a set of pairwise disjoint nonempty subsets of A which cover A -(4) The sets SL + X (L e 2f, x e S ) are called Jines. Two l ines of the same { S^ + X I tX e S}, L e J are ter -med mutually j^arallel. (5) The neutral element of (Fy +•) i s denoted by 0, whe-reas the neutral element of (F \*f O j O i s denoted by 1 . - 420 -i i ) ( F \ { O j / ) i s a loop < 5 \ i i i ) 0 ' X =- X < 0 - 0 for a l l X e F > iv) X-('ty+z)=X'ty+X'% for a l l X, ty,** F (j^jH^jstnttfti^J• . We say that ( F- •+-, • ) i s Jglanar i f v) the map x - » - (a* * X ) +&• X ia a bijection S - V ->S for each pair of distinct (*,<, -& 6 S . Definition 3 . Let fl «CF,+, • ) be a quasifield. Then we construct (Fx F, + )^(F1+)e(F7+)aiiA PttZ)1* {({(x,<y,)\«£* * cux]7 + )\a,eF\v {({*,<ty]\x =0}, + )} -Propoaition 2» Let ft = CF,-(-7 • ) be a quasifield. *&en (PC(5.) i s a f -partition of ^-= CFx F, •+• X Q i 8 pla-nar i f f each pair {{ C x , ^ ) l ^ -=• <<X | , {(*,<y,)\ ^ =r/J * ? J with <x, =^ (l i s generating. Proof. ^ . - Cy, +•)= C-fCx,/yJ I X-*0],-rO i s a group isomorphic to 5^CF, + ) . Next, each ^ ( ^ , - r - ) * ( { C x , ^ ) ^ . ^ ^ x i «fO ;a,eF,is an Abelian 8emigroup with neutral element (0, 0)* By right distributivity, if follows that (a,,Q,<4,) + (-cu,-(a,u,)) = ^(0)0)^ so that 3%, must be a subgroup of &'. Each element (u,0<w) £ C F \ { 0 J ) ^ F belongs to exactly o n e i ^ , a e F; Indeed, (^,or)-«CM',-Zit) holds for a uniquely determined CL e £ F such that CL'U, ~ V. Thus $(0,) i s a partition of ?". Now, we shall show that each pair-fy? ^ $ ? a, e F i s generating: The equation y + r̂  =. F i s valid i f f each pair C*t 7 i^e 6 F x F can be expressed in the form (0,v^ )+ (ti^^au^ ) for some ^,1/^ e F . But AJL *0+AL1 , v** tfj +au,i hold for ^ 1 • ^ » < ? a ^ - ^ ^ • 421 -Furthermore, 1 ^ , f-J f ia generating: F0 + f̂  - F holds i f f each (u^iT^e F x F can be expressed as (u1}0) + +(u27u2) tor oertB±n ui7*cz e F. But u *u^ + u z , nr~ sQ+U2 ±B obviously satisfied for Uf « u - v and u^= V. Finally, { f .̂, r> } 7 a * Ar f i s generating i f f each (u}ir)e F x F can be expreased as (u1la,uf)+(ul74ruz) for certain u1} uz e F. But the equations u » /Uf -f-/^ , /t/-* att-i -f- ^ ^ 2 **ave ft ^^-*lQue^ BoX\xt±on(u^)u1)e F x F i f f there exists an u1 € F such that-aUj f-fru^^Jlru-tr (or, in other words, i f f Jru -tr has a (unique) Inverse image relative to the map x —.y -. a,*,x+-£^ ^ ^#-.#-.)This concludes the proof* DettaiiioaJL. Let P - { C S t ? + ) U 6 J J be a X -partition of a nonzero Abelian group (S7 + ) . Let oC, fi> j V be indices with the same meaning as in Definition 1. Then every element a € S admits a unique expression in the form £ a +1&, where faeS^, ^ir e S . . Let JT be a bijection 4c ~~* ^ 8 U 0 h t h a t «* -f-^-* e S ^ for a l l j c e S ^ * Let ae be a bijection S^ — * ^ \ f / 3 £ such that(ae o7i)c =B where -f c $ = CY -+• tf'e ) o S^ for a fixed eleaant € in S ^ \ Y 0} . Let there be given a multiplication * on ^ s u c h that a - i r ^ e , where ic} * S ^ ^ o CS*, + jrJr). Finally, l e t $,CP) denote the triplet C ^ 7 + 0 • ) * ?yopo§lU9a 3* If P - { ( S n + ) U e J } i s * t: -partition of a nonzero Abelian group C 6 - + ) , then Q(P) ±B a qfuasifield* IP i s planar i f f ^ C P ) la planar. ££&&£• The definition of the multiplication • implies 0*0, ? a,' 0* 0, C'a» a*e «- a for a l l <& <e S^ . The qua-• 422 -•igroup propertiea of ^ \ i0}0 follow from the fo l l o -wing facta: l ) aach element of £c N i°i l i«« in exactly one component of P and 2) ta*d (($* +*)f\ Sac*)* 4 f o r al l * € S and a eS^. Part 2) follows from S^ + S^^ S by Proposition IV. Now verify right diatributivity* Firat, sro, + a and JrAr + 4r belong to Sf 9 so that (fra + a)+ + (rrJr+Arh(na+rr4^+la+lunStyflLth sra, + frfr e S^ and a + + Jtre S^. Thua sra + JrJtr » JT (a + Jlr ) . Therefore JTCL + ta7 JrJtr+ttr e Sa e c . where Jr(a+4r) e S^ and ea + ctre e 5 c -By the definition of JT7 ee 7 • , we conclude that ecu +-c&m -= c Co,-r-^);aa required. Note that the map x —> -ax +Arx (a, 4 <tr) i s a bisection i f f CMcKS^^n ^ + ^ ) ) 3 1 for a l l tifeS, Q.E.D. ^flltftlff l ?- 3>t P - •( C 5^,-H ) I t € 3 J be a partition of a nonzero Abelian group C S , + ) . Let E be the Bet of a l l P -endomorphiama of C S ,-f-) (that i 8 , of a l l endo-morphiama dt of C S 7 + ) for which a e S t s S c ? i e ? ) # Suppose that two compos i t ions + 7 o are defined on E by Caê  + *tz ) a, ~ rt^ a + tt^ a,, (de^o ze^ ) o , « a ^ a ^ ) , where ae--, *€%_ £ k and a e S . Then we shall ca l l the tr ip le t X •= f £.,+,<>) the Jfeejr̂ nel of P . ( 6 > (6) I t i8 well-known that X i s a unitary ring without ze-ro diviaers. The neutral element of (E0 + ) i s the zero endomorphiam 0. ( 0 a ** 0 for a l l CL € S ) , and the neutral element of ( E \ { 0} 7 o ) i 8 the identity auto-morphiam ± (±a = a for a l l a e S ) . Gf#t2],p.l62#If IP i s planar, then X i s a skew-field ( [ l J f p . l 6 7 ) , - 423 -Definition 6. The kejpnel of a quaeifield Q, «CF,+-7 *) i s the subsystem CK,-r-?-) with Ka { ^ I Cx + y*) • A> -=*• s X- & + sy,*/*7 X'(<y,*A>) » Cx *^f) • ^ i • IroDOsition 4. Let there be given a quaeifield Q, = <F^7-)-Then the 9(0,) -endomorphisms of(Fx F, -r-) are precisely the maps C X, ^ ) —* Cx • a . , ^ • a ) with cte K} and the kernel of Q i s isomorphic to the kernel of <P(Q> ) which i s a skew-field. Proof (coincides with t lJ ,pp. 173-175): a) Every map (C6*f Cx ,^) ->Cx-fe , «j,-Jk ) with Jfc<s K \ - f O £ 18 a C^CQ)-automorphism of CFx F., - H ) . Indeed, by a simple cal-culation one obtains ^^CCx . i^)+Cx/ ,^0)»(C^Cx,^)4-(e^X^9-Since (U^C*? a X )-*Cx/fe, (cux Xfc Vfx^oftAfcnd (t%(0,<&)= -CO, ^/rt),it follows that each -fCX,^) / % «- a,x ? or $CX,^) / X - 0 } ? respectively, i s mapped onto i t se l f . b) .Each <P(Q, ) -endomorphism of (Fx F, -f-) has the form Cx, ^ ) —^CX'i&^-A ) for a suitable Jk e K . In-deed, let 0* be a (P(Q> ) -endomorphism distinct from £ • Then tfCx,^)-* &(*>0) + 6 CO,/^) * C^X, <r< .̂) , where f? and G are well-determined maps of F into F . As <Cx,^) 1 ^ » iX ? ie sent into i t s e l f , i t follows that d^s <? , so that 3 * C a x ) « a - e jx because-CCx,^) 1^ . -»<ZX? must be mapped into i t s e l f . Since 6 i s an endo-morphism of (Fx F7 + )? the additivity property ¥ Cx + <#,) * ^ t f x + &'&' «ust be fu l f i l l ed . For Jv * tfd) we obtain <¥(*> 4 ) - X -^ClKx-A^so that C*x)A-^Cax)~#frx--a£</L) ( f irs t kernel property) and Cx-f^)^«X/fV +<y,fo< (second kernel property). * - 424 -Thus He K and ff C*,^)*Cxh>,<yJv) for our A e K * Q) The rest of Proposition 4 ia a corollary of parts a ) -b) and of the footnote C6) . Remark. The study of a l l partitions of Abelian groups the kernel of which i s a skew-field may lead to useful re -su l t s . Proposition 5. There exists a non-planar quasifield. Thus, by Proposition 2, there i s f - p a r t i t i o n which Is not planar* Proof. Let C F7 + <) • ) be a transcendental Dickson nearfield constructed in E4J,§ 4 in such a manner that (F7 +7 * ) i s a f ie ld of rational expreasiona of one inde-terminate *fc over the f ie ld of complex numbers and the new multiplication • i s given by a^Ct) JbjCt) _ UjCt) 9 JfyCt+defr^-cteQ. a^) a^Ct) * Jb^Ct)" a>±Ct) * tr2Ct +deg,d^<ttq, *%) Of0t> *iCt) - x,ci) , / , *i&)\. . XiCt) r- r-v Ct) o f i n t 0 i s n o t 8 U pJ e ( r f c i v e because ^C<1 -X^Cf-M) oc^Cr; It may be noted that Proposition 5 solves the question posed in M. Hall, Projective planes and related topics, Ca-l i fornia Itstitute of Technology 19154, p.48: Indeed, the con-dition Cy) "there i s at mo8t one permutation which displa-ces a l l elements and sends an element Ou into an element fr^ cu " i s not a consequence of 6fc ) n € i s doubly transitive" and ((&) "only the identity permutation f i x e s - 425 -two distinct elements" in a suitable grfrup & of permu*-tations on a set S . fxvnhuw $» Let P«<CŜ -r-) Ue 7 3 *• a ** -partition of a nonzero Abelian group CS, **) an<i $,(&*)' m(ScCo+7') the quaaifield determined as in Proposition 3 . Let 6^^) '*-*&••* +-& be a map of S onto S where a 6 S V f O j - ^ e S . Then in £ ^ F ) the multipli-cation i s associative i f f the following condition holds: CR) For any two fo,,-*?), (a2,\)e(S \iO})x S, the COBK position fy^,« f f c , , ^ ) coincides with some ? C a ^ , Ca„<%)e CS\<0}) x S . Proof* Let the multiplication of %,(P/ be asso-c ia t iv . . Then fc^, . ft^, . ^ a ^ V ^ ' ' * f W i s fu l f i l l ed then 0<2 * # V * ) + 0 / ^ + ij»^*«X + frz holds for a l l X € S ? where o^ ., ^ £V * f., 2., 3 ) have the mea-ning described in CR) • For X -0 one obtains a2^ + +^k m ^3 a n d consequently a2* (a^* x)= a3* X * For x « 1 ? one obtains a2*a^ * 0/3 and consequently CL2*C^ ' «x ) « m(<l2ai)'X' Taua Q(fP) -»*** nave an associative multipli-cation* This completes the proof, It i s well-known (M. Hall, The Theory of Groups, Hew York 1959, p-382) that a) in any associative quaaifield C50 + , ' ) the mentioned maps % * ; ? Ca,&)e (5 \ iO})x S , form a group which acts doubly transitively on 5 and which has the property that only the identity f ixes two distinct elements of S , b) i f (&> o ) i s a group of per-mutations on a set 5 - CAHxi S > i satisfying the two preceding conditions, then there i s an associative quasi-- 426 -field £ « ($?+? '} »«^ **»* *ße coaete <£+<*,, a,€S\{0}, oCeSxS in ( S x S , f - ) where ^ * t o , ^ ) l y * Ä a } > ^ 6 5 \<Ö3 , are precisely the sets { f ^ */*) I <* e S$, R e f e r e n c e s IU J* ANDR.&: Ober nicht-Desargueeache Ebenen mit transi-tiver Translationegruppe, Math.Zeitschr.60  (1954),156-186. 1$ J. ANDR.6: Ober Parallelstrukturen II, Math. Zeit sehr. 76(1961),155-163. [3] R. BAER: Partitionen abelscher Gruppen, Arch.d.Math. 14(1963),73-63. [4] H. KARZEL: Unendliche Dioksonsche Faatkorper, Arch. d.Math.!6(l965),247-256. (Reoeited March 21,1966) 427 -

One generalization of planar partitions of Abelian groups

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One generalization of planar partitions of Abelian groups

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