EXTENDING ABELIAN GROUPS TO RINGS

LYNN M. BATTEN, ROBERT S. COULTER, AND MARIE HENDERSON

Abstract. For any abelian group G and any function f : G → G we define a commutative binary operation or “multiplication” on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p-group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p-group of odd order p2.

1. Introduction The classification of finite simple groups was primarily motivated by the fact that simple groups are the essential building blocks of finite groups. For finite ring theory, the rings of prime power order assume the role of simple groups as the prime objects. Recently, the problem of classifying finite associative rings has received considerable attention, see [5, 6, 20] for example. Meanwhile, work on non-associative rings has tended to concentrate more on construction methods, though there have been some results on the classification problem, such as [11]. In their broadest sense, rings can be viewed as additive groups with an additional bivariate mapping satisfying specific properties used for mul- tiplication. Here we provide a method for constructing non-associative rings based on this viewpoint and then consider the classification prob- lem for the resulting rings, concentrating specifically on the elemen- tary abelian case. In particular, a complete classification is achieved for those rings constructed using our method and elementary abelian groups of odd prime square order. It should be noted that our classifi- cation method does not use standard ring isomorphism, but a weaker equivalence relation which is certainly more natural for our problem and may be significant in the study of non-associative commutative rings in general.

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Soc. 82 (2007) 297-313 to for there assuming L c trace c sho If w implying Adding with Pr d the DO 4.2. Lemma of subtracting they as of It hoices p p 1 e +1 a o w the test = required. follo is ha wn eac of. three = eak binomials: Equiv = cannot c (3) not v is mapping p b h previous e Setting ws that d − for exactly then (3) automorphisms ( c p giv c 4.5. +1 hosen in a equations a c p that c p alence v + + . − with L +1 es one b ertible and (6) d W 1 e d b ( We ) f b = equiv X 2 X T one e in section. EXTENDING c c ( ∈ from implies = r = (5) X p b suc ) ha p to − 2 p +1 have is = F | +1 ) classes d 0, c if − Ω( v | | d (5) p ∗ hoice alen h p = and Ω( + e Ω( equidistributiv aX +1 . and p the or w X that a ∈ a X X X X c e W p w + for + c p c ( t hoices p 2 +1 equating p p c p p ( obtain − a F 2 e other + only , to e +1 +1 of +1 a b p b b cd − = p ∗ − p L and obtain of eac ABELIAN − + b bX shall = = = ) + b eac . + p 1 = (as d | − b suc X X ) DO is X = 2 c d = h No if a for = d 2 d 2 sho is cd X h 2 )( 2 2 . p the 2 of not = a + 2 a h w + ) ) +1 w , other, c no 2 in It b | | p p + a with a p p e − and ws that these binomials. cd ( ( = = +1 c − + . e, v . − taking w c p follo = p equation in c ertible GR ma If b d. Including p − p a 2 p a − X . − there v +1 sho c p ( ) applying a OUPS = ertible. (4) p a p y Multiplying d or ( 2 1)( d ws = +1 binomials, p . + = ) + − p w c p omit +1 − By to +1 p b 0 p the + ev d − that 1) +1 as . are 2 = that p 1) . TO the b en cd +1 − 2 , determining p w 3 c last , see the and Hence, tually . c . W 1) p ell. exactly Lemma c then RINGS +1 t A p a w = e +1 Lemma p . symmetry o t w similar no +1 With w through Ho d = monomial e using o ). w giv there shall = w − equations p 4.3, deal ev es cd Substituting b b c the p 3.1. er, hoices calculation (3) = +1 = sho w are , b with c e w y w n d p and implies w classes As +1 obtain um p e e a p +1 need ha 2 that of − and and t . − b the (5) (5) (6) (4) (3) w v er a  11 It a o 1 e , J. Austral. Math. Soc. 82 (2007) 297-313 remains There 12 T (they suc There one So and (ev remo w So c ( F T from cd T pairs Ov ( there (8) legitimate Raising p c c, (= ( p o r( r( e p F − . 6= erall d en h c c there either b ha or − sho ). ) ) c g As (11) d egin v hoice 0, that whic i ( = 6= v ) are are e 1) ( if f are c, p p are F ws e +1) then ( those w w 2 0 0. cd (7) or = to d X 2( are imply e e coun d w h ) d t (as p / c cd w pairs ) c p This = p of 2 these coun ha eak can when = − it = − to 2 ). o = − − 2( ( ∈ 0). v there 1 d c pairs c follo c c ting, d c W d e X the 2 hoices p hoices. 1) automorphisms suc exclude for F or d t . means or + ∈ 2( ( e − 2 W p pairs c, the Multiplying BA cd p = . ws require c d p h d F p − 1) d hoices whic e d are where if 2 TTEN, Also, − o g p ∗ = d ), = ) th = . w j pairs ( X /cd . for c c ( 1)( η − p er ( hoices and us p W This there 0. Ho 2 p +1) h = − c, the +1 , c b a suc p e ( ). c p ∈ η c d d c Lemma equating ha for w d c 0 COUL / − When p . − − p p 2 ( − ), 2 m p p . and as +1 ev F , c, first So h +1 + then v giv a 0 cd where + F b 3) ust are 1)( p c If e c a for c er, d through c or . ho a η 2 = = = = . ) = there c − TER, the es , + of ∈ ∈ − T Setting η using )( cd d and satisfying osing p p Ov d c 2( therefore d p t cd r( d p eac d d b 4( 2 F w F d 2 4.3 2 − X ossibilit 2 p p = p p ( c p cd +1 ∈ ∈ ∈ when p ∗ follo (7) 1 p 2 o = 0 +1 − p p erall ) 6= p 2 AND − , ( h − − are η +1 ), b ) p − suc ≤ F F F − and yields = . c g (9) (as p b d c + an c 1) 0, p p p ∗ ) − are and i and wing 1) − Assume y η 2 + 2 ( 1) j . c h p = there . p HENDERSON η w p c +1) 1) = w = < y d c c sho 2 a X = − c − − remo , e c + 0 fixed p cd = e hoices yields = 1 (9) 2( +1 0. c/d / . ∈ the 2( 2 − it pairs 2( 3 ha 2 2 relev ha ws . c 0. p , p − p d legitimate F p 6= follo are So 2 = v ( with v , sho v c − p p equations c e − − ∈ a e Lik b e w p 2 c p 0, an − ). y c the will o +1 p 4( 1) − e 2( 1) p F p fixes ws p p ws +1 v ) +1 ewise, c t 1) p ∗ ha then erall. p If p 2 − b 0 pairs = and . pairs conditions: and and equation − c giv c − that ∈ − v = ≤ Conditions T 1 hoices 2 d e b r 1) 1) + F p p . d e ( c i c there i +1 c hoices p c p hoices d for when W = for < c W ) d +1 ≡ p . c c , c . p +1 2 hoices 2 hoices 6= +1 e Equation e there p 2( , j W for or ∈ the the ( d need 6= are 0, p mo p = is 2 e F d c/d for − − for d pairs ∈ ha then p only d case case (10) (10) (11) (12) = d p . left are p 1) (8) (9) (7) +1 for for 1). F +1 v to d 2. ∈ If d 0 e p 2 . . J. Austral. Math. Soc. 82 (2007) 297-313 to a Supp yields a (and familiar 4.3. and w Lemma classes. the Equating Pr t c b same three calculation f w p y ( eak − ∈ X W No o o = determine taking of. next b F ) 2nd e final Equiv d w ose hence fashion automorphisms p equations = ∈ note 2 implies Let set X suc to F kind, a section 4.6. classes, a a a (13) p p p 2 f . f +1 those p alence that th h + + + pairs for ( ( − W X as X that the b b b p = We | | c see and g X Ω( Ω( ) e = = = ) pairs EXTENDING o for + p m the | w w = a = b 2 Ω( who ha ( n X X d c c ust b p a a a, d ers. [15]. e have this + giving um 2 p +1 classes 2 ( (14) X oth f X v X +1 2 2 + + p ∈ equation shall − b − p p e ( g . 2 b ( )). 2 2 X of b 2 b ha − + b b b p p a, F p + p − 2 b 2 The +1 e holds. er Then + of ) p − = = = cd c w − v the g b 2 = d . In satisfied. p p = ) e p e X X g sho = d p p of )(2( 2 g d c 2 W whic 2 +1 X 2 a studied X is of ha ( + 2 X coun + ABELIAN X summary p p − c p a e +1 +1 + a + w p + p p T − p in p olynomial d v +1 exactly p p c +1 DO +1 Therefore also +1 +1 p +1 ∈ e h 2 2 r( g g + + that v α − p p cd cd t + c suc olving either a + + , F + + are + = = p X X for 1) ) d p X p where − ha trinomials. X d X the 2 α X h 2 2 ( ( = + + 2 p w 2 ) ) c c , GR ( ( ) suc v generated 2 +1 2 pairs | | . 2 pairs. p cd α the e d there − . c e = ) − = = resulting − F c Dic 2 2 η | ) OUPS + ha Applying ) a p p X h α rom d = + there + p 2 + . p 2( α d p + p T same v 2 that 2 = kson ) ) and d ( p p 2 ( p e ( η r 2 2 ∈ b c, are p Th p cd 2 ( . − − ( ∈ − Lemma exhausted ( TO a ∈ − c − 1 d p F )) are p X us + F ) 1) a there in p ∗ − F P b as 1) in 1) − . It p p . RINGS follo 2 y p 2( olynomials 2 +1 Lemma . 1) 3 the . 3 or d p p A remains as it DO cd )( 2 ( and 3 6= p ws . 4.3, c − short are ) c w (14) − last p p b − as p trinomials. = p the 1) in +1 d the p c 4.3 for 2 d argumen is hoices p exactly to calculation . ) c pairs. and n hoices of unaltered Ho um follo yields the consider the w b where for ev wing er case (14) (13) (15) (18) (17) (16) The the t 1st for er, In of 13  is a J. Austral. Math. Soc. 82 (2007) 297-313 As Substituting Raising p It almost b So (replace Supp that Multiplying b whenev If F Equating Lik F 14 th rom e p are . follo c ewise, a used, c 2 p If p ose c d is o completely = (16), 2 cd w ws = , er the ( determined ( ( ers) to d g a a cd d a 6= giving 2 cd p sho (18) that c 2 p + + the +1 p , − w ∈ same 0, p +1 ∈ (19) from , b b then ws e in ( g ) ) = (22) then F a ) yields p p 6= F ha to p a p + c + + p 2( o b ag p d v p b w = conditions BA determined (21) (regardless (19) d +1 g with p e b p ( ( y c a implies er +1 p )( a a p p completely 2T ∈ p TTEN, − a ( . +1 g + ( + + ag a . p ag ( + W c F p and using r( 1) d and c p bg Equations +1 b b p p and c 2 = +3 p p e cd b ) ) . 2 p +1 + − + d = = = = ma − Returning COUL pairs ) + subtracting d (20), bg − d + ∈ bg subtracting p c g c c b the for 2 d y +1 p 2 2 2 ) ( of b c ) g p p p +1 p = = = c F + p y assume +3 , ( p b TER, − + + +1 − p ( c +1 v c, ) whether y one c c c ). c, d c a p alues − b + 2 2 2 (16), c c )( +1 ( 2 d p p p c ( 2 2 and + a d case − b + d 2 a − − A ). + + AND as and p + 2 + to 2 obtains c d p g ( g 2 ( − d c p c d d p ab similar from of b d d bg c It +1 (17) (16) − in 2 d 2 2 2 ) from + p p p p w p p c 2 , p − HENDERSON d d +1 +1 ) a ( 6= 2 cd − ( remains + + e , the and a c ) ag + + d p p whic − = and 0. bg = and m (20) d d d 2 g the = p 2T b ) itself 2 2 ) ) ust c ) . case 0 2 0 + coun Ob that and + + 2 = . . = r( 0 h (18), − (18) bg equation sho g g cd 0 viously 0 or exclude. in to p p . d ) w ting f . ag c cd ) a 2 p ws = e . ( ) turn p not). d no sho w p X p +1 obtain + 0 + + e + ) w . argumen w g ha 6= bg g g = 2 generate implies cd c c v that p p b These (and X e d. It p d. p . +1 c 2 2 p follo − a − t holds their d (20) (21) (22) (19) b and can 2 are X ws is. ∈ 2 J. Austral. Math. Soc. 82 (2007) 297-313 X Pr o is p of can X X of Recall previous Th the for Theorem 4.4. already v olynomial). 2 2 2 er easily It o w us p , . DO of. all six X eak b − F A remains There e a p p DO X +1 2 That |G p equal complete classes p . excluded. +1 automorphisms established olynomials, 2 three , | Let , X = 4.7. f p = are 5 p olynomials the ( +1 to for f p W X b 1 ( sections, giv p + ( p EXTENDING Under e p sho ) +1 X six a 6 X − | | | | | | need = en C C C C C C Hence ) − set fixed w 2 implies 1)( ( ( ( ( ( ( = p , X from f f f f f f 1 in X olynomials and that 6 5 1 2 4 3 X we 2 of p ) ) ) ) ) ) to DO 2 p o | | | | | | 2 the p w prime (as 2 v − ak p ======− | equiv , − sho e er Ω( noting 6 these these f X 2 p c     ha   − 2 ABELIAN e 1). previous determined X X p ( F olynomials quivalenc w 2 +1 X p p p p p p v 1 , p 2 p 2 2 2 2 2 2 p 2 e X p alence +1 ) = . 2 2 2 2 2 2 − − − − − − By six ar > = giv + = that 2 p X + 1 1 1 1 1 1 i e − 2. =1 g 6 X d en classes       Lemma X X 2 p describ GR three p +1 e, X ( (2 ( (2 ( ( | no +1 p C alw 2 p p p p classes. +1 OUPS p p p 4 2 2 4 in ther o ( and , +1 whic 3 t f v − + − − + f a + w i er − the 3 con + ys ) sections e o p ( p p 2 2) | e 4.2 X X d . X ) ) p 2 2 f F TO h ar form ) describ p 3 tain 6 2 previous ) . p 2 by ) ( ) + 2 e are and X = | and W RINGS six (w = p ) ulae the X e all 2 the e do ) 2( = need e X the p e quivalenc +1 exclude p DO six X differen 2 for indeed p − three + cases + 2 results p to X 1) p g p the + olynomials X olynomials 2 2 sho , . g sections) p f t w the accoun +1 e n X 4 e w classes um ( classes of + p X +1 ha that zero X ) b the v er 15  = + 2 e t . J. Austral. Math. Soc. 82 (2007) 297-313 Summing p seen 16 us p is Theorem as not nomials ha Conjecture rings Pr W if found. isomorphic mial p b the 2 olynomial olynomials e e f not. W Since A The v o + required. to p of. e ( pro planar. isomorphism X e erm 3 p that ∆ b mak of p end olynomial ) een It f metho So ceed + ≡ ( utation order t whic X w follo e 6 this giv the w L used o , 5.1. in the rings if a e F 1 to necessarily rings h ( es ) or d ws ∆ ∆ ∆ ∆ need L 4.8. a p article p construction is are sho 2 2 used X follo f f f f 2 class i > A to the p ( 6 5 4 3 =1 6 f from a X ( ( ( ( for BA problem olynomials { X X X X 3 DO w are F construct planar only p L o | )) wing TTEN, C remaining , , , , p and v erm et ab is b an 2 a a a a that er mo ( , y ) ) ) ) [7 w p f 5. a + o y p olynomial i = = = = describ applying test eakly , utation F v ) p planar , d conjecture. | e o Theorem Planar q ? 2 an o eac COUL (2 2(( 2(( aX = = pro dd ( f v + for X is metho } a er the pro p  of a aX h p p 3 o p vided prime gener planar 6 isomorphic p 2 class dd p w p + e + F TER, of − function, F − − 2 p + jectiv planarit eak ) p semifield p our g ( p 2 2 − olynomial 1 f d 2 these a a prime. X . a 5 − Functions 3.3] ) p ate ) for 1 ∈ represen AND go yields X By ) X p if isomorphism + aX results  , if e . F d p o p wher p 2 all (2 planes, that p for d Computational − [7, difference + y a by ) 2 = HENDERSON or p [ ) class if X ( of planes. (2 X 4 a at a 3 Then Theorem ev e p ev + o ] tativ . p they to a olynomials ∈ X the v L is ery least − ery + 2 er 2 1 F classify see p represen , planar a g p ∗ is 2 class L es, F ) 2 the p are a a + ma X . 2 p q p olynomial planar six [9]. . ) olynomials ∈ ∈ No ) it 2) X 2.3], y Planar isomorphic, numb G represen F is non-isomorphic ) those w if b All . . tativ evidence q ∗ f e , easy and a and either adapted ∈ the p er planar +1 es DO F functions in and to − tativ of p p X 2 cannot can a olyno- [ ( p sho X ev p leads class non- +1 a only it oly- DO p ] ery es. for b w  + is is is e J. Austral. Math. Soc. 82 (2007) 297-313 of sarguesian an are plane L whic als (2 p aX ∆ [14] tributiv [12] [13] [11] [10] 2 [1] [6] [2] [5] [4] [7] [3] [9] [8] a erm 1 f a It ) y order 5 ◦ p p p if ( no D.E. I.N. E. E.G. L.E. L. B. R.S. S.R. P P Algebra So Algebra (1964), or angew. class Math. 28 10 Berlin, 217. +1 a + h X X has + utation . . and der of c. ∈ Carlitz, ( Dem , is Dem Kleinfeld, Corbas 2 (2000), (1909), a other = g a e, Herstein, Coulter Ca 75 ◦ Dic F p Kn Go p a II ) order planar b n + 2 p ∗ 3 ) L only − , , , , there b 2 een 1968, vior, Math. are 119–136. b 2 X plane, uth, (1953), o kson, , R 231 (A.A. o Des. 2 Equivalenc 2 is Invariant , (1954), o 2( daire wski Math. a ings wski, p b mo Invariantive planar p 3335–3349. 123–158. and ) desarguesian a oth kno + X nev p olynomial. Equivalenc and if p Finite (2000), reprin EXTENDING Co A 2 and Gener are d T 225 of and + Alb . and (2 X is opics 405–427. G.D. see Finite wn er ( des Z. 382–413. char ∆ R.W. or X a a p not ert, p f the (1967), ) T.G. der e ted 103 DO ev − Y. semifields 4 p + p 2 al [7 Cryptogr. acterization for − ( 2 677–690. classes erm in Williams, ory ery X X = , a Zhou, − Matthews, the ed.), g p 1997. e the Ge 1 (1968), Theorem 5 , a A p Ostrom, p some classes X a is. ory − p utation Part of lgebr c olynomials erm ory ometries planar ) 191–202. ) Hence and hoices ABELIAN 2T ) X MAA, References and of systems A Since of with and of utation a lternative r 10 II. sets , = 239–258. time, ( R mo that of e a 2nd Planes quations of Planar ∆ ings L ) (1997), pr , Pren DO functions of dular a 2 X o 5.2]. p L of f X . c the oje Springer-V (2 5 oly of al ed., 1 2 ( the As a GR see p , p of X ctive tice whic p rings olynomials + p L e is − Cayley onomials. of rings ∈ quations functions invariants olynomial Th , olynomial OUPS 2 John or 167–184. in the a class g [14 X the or F ∈ Hall, der ) )T planes , over us h a p ∗ der J. are G 2 has of ], finite r( erlag, trace describ Wiley . p Numb Theorem only TO Algebra suc that smal C X n 5 1963, a in p and , , p ( over Part with finite ), erm h J. 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