scieee AI-readable full text Open interactive document viewer

Distribution of low dimensional Malcev algebras over finite fields into isomorphism and isotopism classes

Falcón Ganfornina, Óscar Jesús; Falcón Ganfornina, Raúl Manuel; Núñez Valdés, Juan

Abstract

This paper addresses the distribution into isomorphism and isotopism classes of the set Mn,p of n-dimensional Malcev algebras over the finite field Fp, with p prime. This distribution is explicitly determined for dimension 2 and for the sets M3,p, with p<=7.

Full text

0123445678924741762727414734 272627754634259 67364734757867441678  ! "#$%&#'(%#)*)+,)-./#01*$#)*2,32,4152,61'&2$)51& 7*#%171,/$#*%)#$)0)&89#$0 2*/#$)%)8#$0 4,2$$1$ : ;$42&<=>2,4:)* ? @A2:(,3=>2,4:)* B2*/<(2*C:(D*1E ? ?F41472G4241 75H2228 I76J4196 24J64 BF41472K6454639I76J4196 24J64 LMNOPQRSTUVWXYWZ[\W]SS^_T`aWVWXYWZ[bT^_T`cZdWXe_T[bT^_T fghijkli mnOQoNoLpNqqpLQQLQrnLqOQrpOstrOuvOvruOQuMuponOQMNvqOQuruoOQMwPNQQLQuxrnL QLry z{|ux}~qOMLvQOuvNPNPwLNPLspNQuLprnLvOrLLPq | `OrnopOMLmnOQ qOQrpOstrOuvOQLoPOwOrPqLrLpMOvLqxupqOMLvQOuvNvqxuprnLQLrQy {| `Orn 4 215934J8417644539963627 F  *%&)/(4%#)* oNprOwtPNproLuxvuv~NQQuwONrOLNPLspNQwuvQOQrQux278648419`nOwnLpL OvrpuqtwLqsNPwLNQrnLrNvLvrNPLspNQuxNvNPrOwutxNvPuuoQNrLp`Ov` rnLQLNPLspNQLpLwNPPLq34J8419sNPLpLMNpNsPLopusPLM pLPNrLqru rnLMOQrnNruxrnLOpqOQrpOstrOuvOvruOQuMuponOQMwPNQQLQ vrnLoNoLpLwuvQOqLprnLQLry ¡¢£ux}~qOMLvQOuvNPNPwLNPLspNQuLprnLvOrL LPq £ `OrnopOML¤LQrtqrnLqOQrpOstrOuvuxQtwnNPLspNQ`vuruvPOvruOQuMuponOQM wPNQQLQ`nOwnOQrnLtQtNPwpOrLpOuv`strNPQuOvruOQuruoOQMwPNQQLQrOQNvuLPwuvrpOstrOuv uvrnOQQts¥Lwr mnLQrptwrtpLuxrnLoNoLpOQNQxuPPuQvLwrOuvLLouQLQuMLopLPOMOvNp wuvwLorQNvqpLQtPrQuvNPwLNPLspNQ vLwrOuvLQnu rnLqOQrpOstrOuvux}~ qOMLvQOuvNPNPwLNPLspNQOvy ¡¢£ ¦QMNOvpLQtPrQ`LopuLrnNrLLpoNOpuxru~ qOMLvQOuvNPvuv~sLPONvNPwLNPLspNQNpLOQuMuponOwNvqnLvwL`OQuruoOw§tprnLp`L opuLrnNrNPwLNPLspNQOvrnLQLrQy ¨¢B `y ¨¢¨ `y ¨¢©Nvqy ¨¢ªwNvpLQoLwrOLPsL « ¬®®¯° ±¯²³´µ¶·¸·¹¸º»¶¸¼¼½¾¸½ ¿ÀÁÂÃÄÅÆÃÇÈÃÃÉÊËÅ ÌÍÎÏÎÐÑÒÌÍÏÓÔÕÌÍÎÖÎÑÌÍÏרÓÍÍÙÍÎÚÛÓØ×ÙÜÓØÝÙÞÐÓÍ qOQrpOstrLqOvruQLLv`vOvL`NvqOQuMuponOQMwPNQQLQvNPPrnLwNQLQ`rnLvtMsLpux OQuruoOQMwPNQQLQOQxutp ßurLrnNr`xuppLNQuvQuxPLvrn`rnLopuuxQuxrnLpLQtPrQusrNOvLqNpLuMOrrLqOvrnOQ LrLvqLqNsQrpNwrmnLNpLQnuvOvrnLxtPPoNoLp à á&1,#0#*2$ vrnOQQLwrOuvLLouQLQuMLsNQOwwuvwLorQNvqpLQtPrQuvNPwLNPLspNQtQLqrnputn~ utrrnLoNoLp¤LpLxLprurnLNprOwPLuxNPLxupMupLqLrNOPQNsutrrnOQruoOw 34J841âuLpNLPqãOQNã~LwrupQoNwLLvquLqOrnNsOPOvLNpopuqtwr QtwnrnNr ä BåæçxupNPPäèâç éê äëåìëäçxupNPPäçëèâç éê äëíäîåéäëíîêäïéëîíäêäïéîäíäêëçxupNPPäçëçîè⦠éê mnLPNQrwuvqOrOuvOQLðtONPLvrrurnL34J65476 ñéäçëçîêåòéäçëçîêäìòéäçëçäîêåæçxupNPPäçëçîèâç éóê nLpLòOQrnL3267qLvLqNQòéäçëçîêåéäëêîïéëîêäïéîäêë`xupNPPäçëçîèâ xòéäçëçîêåæxupNPPäçëçîèâ `rnLvrnOQOQNOLNPLspN ôOLvNsNQOQõö ? 禦¦çö ¡ ÷uxrnLNPwLNPLspNâ`rnOQOQwnNpNwrLpOøLqsOrQ91314 327979OrnpLQoLwrrurnNrsNQOQ`rnNrOQ`srnLwuLùwOLvrQú û üýèãuxrnLopuqtwrQ ö ü ö ýå ¡ þ ûÿ? ú û üý ö ûxupNPP  012 }¦ é 3 ê mnL34716 4 41uxNQtsQLr 5 uxrnLNPwLNPLspNâOQrnLQLr 6 Lv âé 5 êåõäèâ 7 äëå æçxupNPPëè 5 ÷v654uxâOQNvLwrupQtsQoNwL 58 âQtwnrnNr 5 âåõäëRäè 5 çëè â÷ 85 rOQwNPPLqK467Ox 5 âåõæ÷mnL34741uxâOQrnLOqLNP 9 éâêå 6 Lv âéâêv oNprOwtPNp`OxqOM 9 éâêå}`rnLvrnLNPwLNPLspNâOQsLPONvmnLvLrpLQtPrxuPPuQ QrpNOnrxupNpqxpuMrnLqLvOrOuvuxNPwLNPLspN  k  4â47}5647962784147524566674112539 69492327566279    75    !qOM 9 éâê  }ì47â6934J841!  mu}~qOMLvQOuvNPNPwLNPLspNQâNvqâ  NpLQNOqrusL692263OxrnLpLLOQrrnpLL pLtPNpPOvLNprpNvQxupMNrOuvQ  `  Nvq  sLrLLvrnLMQtwnrnNr  éäê  éëêå  éäëê xupNPP äçëè⦠éê « ¬®®¯° ±¯²³´µ¶·¸·¹¸º»¶¸¼¼½¾¸½ ¿ÀÁÂÃÄÅÉÃÇÈÃÃÉÊËÅ   Í×ÓÐ  ÓØ×  ÎÔ  Ó   ØÛ  ÓØ×  ÎÔ    ! ÔÙ " mnLrpOoPLé  ç  ç  êOQQNOqrusLNv692269sLrLLvâNvqâ  rOQqLvurLqNQâ # â   x  å  å  `rnLv  OQNv6922169sLrLLvâNvqâ  NvqrnLQLNPwLNPLspNQNpL QNOqrusL6922163`nOwnOQqLvurLqNQâ $ åâ  ôOLvNsNQOQõö ? 禦¦çö ¡ ÷uxâNvq NsNQOQõö  ? 禦¦çö  ¡ ÷uxâ  `LNwnpLtPNpPOvLNpNooPOwNrOuv  sLrLLvrnLQLNPwLNPLspNQ OQtvOðtLPpLPNrLqruNQðtNpLMNrpOñ % å é  üý êQtwnrnNr  éö ü êå & ¡ ýÿ?  üý ö  ý `xupNPP 0 èõ禦¦ç}÷mnLvLrpLQtPrnuPqQ  k  69J41645 !922699234J841914941J44564796272461347419! !H4}56479627K46734J841697269226327 24134J841! !G6J472966J4674841} ' 4J41 }5647962734J8412969õö ? 禦¦çö ¡ ÷ 29434741956479627}ì696922163227424226782727 692216334J8419    H27454416745 41253ö ? ö Båö ? !    H27454416745 41253ö ? ö Båö ¨ !  vrnLvLrQLwrOuvLQLLrnNrrnLQLry ¡¢£ux}~qOMLvQOuvNPNPwLNPLspNQuLprnL vOrLLPq £NvqrnLOpOQuMuponOQM wPNQQLQwNvsLOqLvrOLqOrnNoNOpuxNPLspNOwQLrQ qLvLqsøLpu~qOMLvQOuvNPpNqOwNPOqLNPQ§puM vuuv`LqLvurLs £  ( rnLpOvux ouPvuMONPQOvrnLQLruxNpONsPLQ ( åõ ( ? 禦¦ç ( ¡ ÷uLprnLvOrLLPq £ ôOLvNvOqLNP ) Ov £  ( `rnL8416394qLvLqs ) OQrnLQLr * é ) êåõ + è ¡ £ R  é + êåæxupNPP  è ) ÷ xrnLOqLNP ) OQøLpu~qOMLvQOuvNP`rnLvrnLvtMsLpuxouOvrQuxrnLNPLspNOwQLr * é ) êOQPLQQ rnNvrnL , ptPPqOMLvQOuvuxrnLðturOLvrpOv £  (  -) mnLLðtNPOrnuPqQnLvrnLOqLNPOQ pNqOwNP`OvnOwnwNQLrnLqOMLvQOuvwuOvwOqLQOrnrnLvtMsLpuxQrNvqNpqMuvuMONPQux rnLOqLNP )  . / ,61'&2#4$1%$&1,2%1/%)y }ç vrnLpQroPNwL`PLrtQwuvQOqLprnLwNQL}å 0 Lpru~qOMLvQOuvNPNPLspNuLpNLPq ã`vurvLwLQQNpOPvOrL`OrnsNQOQõö ? çö B ÷NvqLvquLqOrnNsOPOvLNpopuqtwrQNrOQxOv wuvqOrOuvQéêNvqéêOQtvOðtLPqLrLpMOvLqsNoNOpuxQrptwrtpLwuvQrNvrQ + ç 1 èãQtwn rnNrö ? ö Bå + ö ?ï 1 ö B §puMLMMNrnOQOQNNPwLNPLspN`nOwnLqLvurLNQâ 23  mnLvLrpLQtPrQnuPq  k 4 4â 23 75â 2 5 3 5 42692216325647962734J84192J41 45ã!ñ % å é  üý ê6949 6 1416 7 1445276922169  44472 841947669J41645 « ¬®®¯° ±¯²³´µ¶·¸·¹¸º»¶¸¼¼½¾¸½ ¿ÀÁÂÃÄÅÊÃÇÈÃÃÉÊËÅ ÌÍÎÏÎÐÑÒÌÍÏÓÔÕÌÍÎÖÎÑÌÍÏרÓÍÍÙÍÎÚÛÓØ×ÙÜÓØÝÙÞÐÓÍ ! + ??ï 1 ?Bå +  qLréñ % ê! ! + B?ï 1 BBå 1  qLréñ % ê! 8 j 9:9 h ; i ;9<4 J41 612256479627727K46734J84192J4145 ã146922163! ¤LxuwtQvuuvrnLwNQL} ' §OpQrP`LNvNPøLrnLopusPLMuxqLrLpMOvOvnLv Nv}~qOMLvQOuvNPNPLspNuLprnLvOrLLPq £ `OrnopOML`OQNNPwLNPLspNmurnOQ Lvq`LwuvQOqLprnLQLruxé} ¨ì} B ê - NpONsPLQúåõú û üý R 0 ç 2 ç = èõ禦¦ç}÷ç 012 ÷NvqrnL }~qOMLvQOuvNPNPLspNâuxsNQOQõö ? 禦¦çö ¡ ÷uLprnLouPvuMONPpOv £ úLvquLqOrn NsOPOvLNpopuqtwrQNrOQxOvwuvqOrOuvQéêNvqéêNvqnuQLQrptwrtpLwuvQrNvrQwuOvwOqL OrnrnLNpONsPLQú û üý ôOLväçëçîèâNvq > èõ禦¦ç}÷`PLro ?@AB sLrnLouPvuMONPOv ãú rnNrwuvQrOrtrLQrnLwuLùwOLvruxö B OvrnLNPwLOqLvrOrñéäçëçîêåæmnLvLr pLQtPrnuPqQ CD9 j  44 H494y ¡¢£ 374654764564841639454745 4 4 412 564796271563654 ) ¡¢£å E o ?@AB Räçëçîèâç > èõ禦¦ç}÷ F ï E éú û üý ê £ìú û üý R 01 2 }ç = }÷ FG  £ ú! H 496549 7 y¡¢£ 7 åqOM I é £ úê -) ¡¢£ ê!   7 y¨¢£ 7 7 y¨¢£ - $ å 7 7 y¨¢£ -#7  æ  ó  `ó  ó 3 ` 3  ó  ó` 33  ó mNsPLR J OQrpOstrOuvuxNPwLNPLspNQuxy ¨¢£ `xupNPPopOML ¤LnNLOMoPLMLvrLqmnLupLMOvrnLuoLvwuMotrLpNPLspNQQrLMxupouPvuMONP wuMotrNrOuvQ K ÌÔÝ  ØÓÐOvupqLpruqLrLpMOvLOvmNsPLrnLvtMsLpuxNPwLNPLspNQ uxy ¨¢£OrnpLQoLwrruNOLvsNQOQõö ? çö B çö ¨ ÷`xupNPPopOMLmnLqOQrpOstrOuvux rnLQLNPLspNQOvruOQuMuponOQM wPNQQLQ`nOwnOQLoPOwOrPLouQLqOvmnLupLM  3 `nNQ NPQusLLvqLrLpMOvLqstQOvrnOQouPvuMONPMLrnuqvrnOQpLQoLwr`OLvruNPwL NPLspNQâNvqâ  Ovy ¡¢£Ov £ `LvLLqruvunLvNpLtPNpPOvLNprpNvQxupMNrOuv  sLrLLvrnLQLNPLspNQOQNvOQuMuponOQMmurnOQLvq`LwuvQOqLprnLQLrux} BNpONsPLQ  åõ  ?? 禦¦ç  ¡¡ ÷`nOwnLwuvQOqLprusLrnLLvrpOLQuxrnLwuppLQouvqOvMNrpOñ % ux « ¬®®¯° ±¯²³´µ¶·¸·¹¸º»¶¸¼¼½¾¸½ ¿ÀÁÂÃÄÅ L ÃÇÈÃÃÉÊËÅ   Í×ÓÐ  ÓØ×  ÎÔ  Ó   ØÛ  ÓØ×  ÎÔ    ! ÔÙ " utpOQuMuponOQM  Lro üý B sLrnLouPvuMONPOvrnLouPvuMONPpOv £   rnNrwuvQrOrtrLQ rnLwuLùwOLvruxö B OvrnLOqLvrOr  éö ü ö ý êå  éö ü ê  éö ý êux 6 uvqOrOuvéê`xupNPP 0 ç 2 ç > è õ禦¦ç}÷mnLvLrpLQtPrnuPqQ CD9 j  4M H4942692216994447234J8419âçâ  èy ¡¢£ 37 4654764564841639454745 4 4 412564796271563654 ) â¢â 5 å E o üý B R 0 ç 2 ç > èõ禦¦ç}÷ F ï E £ üýì  üý R 0 ç 2 èõ禦¦ç}÷ F ï E qLréñ % ê £ N ?ì FG  £   !H4 315676 2699469qOM I é £   ê -) â¢â 5 ê!  LrtQqLvurLsâ ? ¢ @ ¢ A rnLNPwLNPLspNuxsNQOQõö ? çö B çö ¨ ÷Ovy ¨¢£ QtwnrnNr ö ? ö Båä`ö ? ö ¨åëNvqö B ö ¨åî¤LLouQLOvrnLvLrrupLQtPrQrnLLoPOwOrqOQrpOstrOuv uxy ¨¢B `y ¨¢¨Nvqy ¨¢©OvruOQuMuponOQMNvqOQuruoOQMwPNQQLQ CD9 j  4O H4144 7 69 !94J476922169 3994967y ¨¢B  â P ¢ P ¢ P çâ P ¢ P ¢ Q R çâ P ¢ P ¢ Q S çâ P ¢ Q T ¢ Q R çâ P ¢ Q T ¢ Q R UQ T çâ P ¢ Q R ¢ Q T çâ Q S ¢ Q T ¢ Q R ¦ !767469221693994967y ¨¢¨  â P ¢ P ¢ P çâ P ¢ P ¢ Q R çâ P ¢ P ¢ Q S çâ P ¢ Q T ¢ Q R çâ P ¢ Q T ¢ Q R UQ T çâ P ¢ Q T ¢B Q R çâ P ¢ Q T ¢B Q R UQ T çâ P ¢ Q R ¢ Q T çâ Q S ¢ Q T ¢ Q R ¦ !69221693994967y ¨¢©  â P ¢ P ¢ P çâ P ¢ P ¢ Q R çâ P ¢ P ¢ Q S çâ P ¢ Q T ¢ Q R çâ P ¢ Q T ¢ Q R UQ T çâ P ¢ Q T ¢ Q R U B Q T çâ P ¢ Q T ¢B Q R ç â P ¢ Q T ¢B Q R UQ T çâ P ¢ Q T ¢B Q R U B Q T çâ P ¢ Q R ¢ Q T çâ Q S ¢ Q T ¢ Q R ¦ V !69221693994967y ¨¢ª  â P ¢ P ¢ P çâ P ¢ P ¢ Q R çâ P ¢ P ¢ Q S çâ P ¢ Q T ¢ Q R çâ P ¢ Q T ¢ Q R UQ T çâ P ¢ Q T ¢ Q R U B Q T çâ P ¢ Q T ¢ Q R U ¨ Q T ç â P ¢ Q T ¢¨ Q R çâ P ¢ Q T ¢¨ Q R UQ T çâ P ¢ Q T ¢¨ Q R U B Q T çâ P ¢ Q T ¢¨ Q R U ¨ Q T çâ P ¢ Q R ¢ Q T çâ Q S ¢ Q T ¢ Q R ¦  CD9 j  4W H4144 7 6921692269 3994967y ¨¢£ 21164 â P ¢ P ¢ P çâ Q R ¢ P ¢ P çâ Q S ¢ Q T ¢ P çâ Q S ¢ Q T ¢ Q R ¦  « ¬®®¯° ±¯²³´µ¶·¸·¹¸º»¶¸¼¼½¾¸½ ¿ÀÁÂÃÄÅÄÃÇÈÃÃÉÊËÅ ÌÍÎÏÎÐÑÒÌÍÏÓÔÕÌÍÎÖÎÑÌÍÏרÓÍÍÙÍÎÚÛÓØ×ÙÜÓØÝÙÞÐÓÍ X >(&%91&$%(/#1$ J LNPOvOrnrnLQLry ¡¢£ ux}~qOMLvQOuvNPNPwLNPLspNQuLprnLvOrLLPq £ `L nNLLvtMLpNrLqNvqwPNQQOLqOvruOQuMuponOQMNvqOQuruoOQMwPNQQLQrnLLPLMLvrQuxrnL QLry B¢£ `xupNvopOML`NvqrnuQLuxrnLQLrQy ¨¢£ `Orn§tprnLpQrtqOLQOvrnL Qts¥LwrNpLsLOvNvNPøLqruqLNPOrnrnLwNQL ' NvqNPQuruwnNpNwrLpOøLrnLqOQrOvwr opuoLprOLQuxrnLNPLspNOwQLrQpLPNrLqrurnL9 Y 4427uxNNPwLNPLspN`nOwnOQNvL wuvwLorrnNrLOQnruOvrpuqtwLNQNruuPNvqrnNrLNpLvuNvNPøOv ¤LNpLNPQurpOvruNQQuwONrLNwuMsOvNrupONPQrptwrtpLOrnLLpNPwLNPLspN` OvrnLQNMLNNQOvórutPqNPPutQrutQLopuoLprOLQuxrnLQLQrptwrtpLQruMNL LNQOLprnLQrtquxrnLNPLspNQ / 4 Z *)-,1/6101*%$ mnOQupnNQsLLvoNprONPPQtoouprLqs [  J vqNPtQONvputo§ \ ~æ` [  J  vqNPtQONvputo§ \ ~`mæ~óæó 33 ~ [[ pu¥LwrNvq§ 0J0]  A1+1&1*41$  ^_ Ù× ` ÙÐ a Û a ÐÙ  ÙØ ab ÚÌÍÖÙÐ  ÓÔÕ cK ×Ò de ÔÙÏÓÔÔ`6781 V  K 32418419 94 212 726326279`nrroR ff QOvtPNptvO~PqL` æ 3   gh ÛÓØ×ÙÜ`K7 63229`NrsupvOß 4W R i é 33 ê 3  j3   ggK ÓÝØÙ`34J8419`mpNvQMLpNrnuw k Réêó j ó 3i  ó gl ÓÐÐÌÓ " Î m Û  ÙÐÔ  ÓÔÕÙ " ÓÔÕ    ! ÔÙ " `2672169131499923645 6648419276456479627`OvLNpPLspNooP 4no éææóêó j  « ¬®®¯° ±¯²³´µ¶·¸·¹¸º»¶¸¼¼½¾¸½ ¿ÀÁÂÃÄÅ p ÃÇÈÃÃÉÊËÅ View publication statsView publication stats