Isogroups and isosubgroups

RACSAM Re . R. Acad. Cien. Se ie A. Ma . VOL.97 (1), 2003, pp. 1–12 ´ Algeb a / Algeb a Isog oups and isosubg oups Ra´ul M. Falc´ on and Juan N´u˜ nez Abs ac . The main goal o his pape is o gi e a ma hema ical ounda ion, se ious and consis en , o some pa s o San illi’s iso heo y. We s udy he iso opic li ings o g oups and subg oups and we also deal wi h he di e ences be ween an isosubg oup and a subg oup o an isog oup. Finally, some links be ween his iso heo y and he s anda d g oups heo y, e e ed o ep esen a ion and equi alence ela ions among g oups a e shown. Isog upos e isosubg upos Resumen. El p incipal obje i o de es e a ´ ıculo es p opo ciona un undamen o ma em´ a ico, consis en e y igu oso, a de e minadas pa es de la iso eo ´ ıa de San illi. En ´ el se ealiza el le an amien o iso ´ opico de los g upos y subg upos, es udi´ andose asimismo la di e encia en e un isosubg upo y un subg upo de un isog upo. Se mues an inalmen e algunas elaciones en e es a iso eo ´ ıa y la eo ´ ıa s anda d de g upos, e e en es a los emas de ep esen aci´ on de g upos y de elaciones de equi alencia en e g upos. 1.In oduc ion In 1978, he I alian-Ame ican heo e ical physicis and ma hema ic Rugge o Ma ia San illi p oposes a gene aliza ion o con en ional Lie’s heo y by using he concep o iso opy (in he G eek sense o being ”axiom-p ese ing”, also called iso opic li ing), which implies he o igin o he ac ually known like San- illi’s iso heo y (see [2]). To do his, he ex ends he basic uni I=(+1,diag(+1, ..., +1), ...) o he ini ial s uc u e o a gene alized uni b I=b I(x, • x,•• x, ..., µ, τ, ...), called isouni , which depends on he coo dina e x and hei de i a i es, on he densi y µ, on he empe a u e τand, in gene al, on any magni ude o he physic en i onmen o he sys em in which we a e. By using i , San illi does a s ep-by-s ep gene aliza ion o he mo e impo an ma hema ical s uc u es, ob aining o he new ones, cha ac e ized by he ac o ha ing he same p ope ies as he ini ial ones, while he new uni s sa is y mo e gene al condi ions han he e i ied by he ini ial ones. San illi gi es he name o ma hema ical isos uc u es o hese new s uc u es. In his way, he s udies isog oups,iso ings,iso ields,iso ec o spaces and isoalgeb as (see [3], [4], [5] and [11], o ins ance). I allowed him o ge in a as way some de elopmen o physical applica ions, p incipally in Quan um Mechanics and Dynamical P oblems o pa icles and an ipa icles. San illi’s iso opies allow o map any gi en and ixed linea , local and canonical s uc u e in o i s mos gene al possible non-linea , non-local P esen ado po Jes´ us Ilde onso D´ ıaz. Recibido: 6 de Feb e o de 2002. Acep ado: 30 de Oc ub e de 2002. Palab as cla e / Keywo ds: Iso heo y, Lie-San illi, isog oup, isosubg oup Ma hema ics Subjec Classi ica ions: 17B99. c °2003 Real Academia de Ciencias, Espa˜ na. 1 R. M. Falc´ on and J. N´ u˜ nez and non-canonical o ms which a e capable o econs uc ing linea i y, locali y and canonici y in ce ain gene alized isospaces and iso ields wi hin he ixed ine ial coo dina es o he obse e . Howe e , in he las yea s, San illi has ound some ma hema ical inconsis encies in his ea ly o mula ion o he iso heo y. Due o i , San illi and o he ma hema icians ha e s udied he iso opic li ings o Func ional Analysis and Di e en ial Calculus (see [1] and [6]). I has allowed o ge some impo an applica ions in Physics (see [7], [8], [9] and [10], o ins ance). So, as San illi’s iso heo y needs e en a consis en ma hema ical ounda ion, San illi himsel has p o- posed se e al subjec s o esea ch o he in e na ional ma hema ical-scien i ic communi y. One o hem consis s on p o ing he exis ence o isos uc u es co esponding o he li ing o s uc u es al eady known, al hough hey do no ha e a p ac ical applica ion in Physic. San illi hinks ha i would be good o con ince he scien i ic abou he ele ance o his esea ch, which would gi e bigge consis ence and eliance o his iso heo y. In his pape , we y o pa ially esponse o San illi’s pe i ion, by s udying a possible li ing o he simples algeb aic s uc u e: he g oup s uc u e. Howe e , as he e is al eady some s udies abou i (see [11]), we comple e hem, gi e also some examples and show how he subg oups can be iso opically li ed by using he San illi’s model o cons uc isop oduc s. Ge ing his las ques ion is he main goal o his pape . To do his, we p e iously gi e in Sec ion 2 some basic de ini ions ela ed o iso opic li ings. Sec ion 3 is de o ed o he s udy o isog oups. Nex , in Sec ion 4, we ob ain Theo em 2, which assu es he cons uc ion o an isosubg oup, gi ing some examples, oo. We also dis inguish be ween subg oups and isosubg oups o an isog oup. Las wo sec ions a e de o ed o s udy some applica ions o San illi’s iso heo y: ep esen a ion and equi alence ela ions among isog oups, espec i ely, and some links among hese concep s and he s anda d g oups heo y. 2.P elimina ies Remembe ha o a gi en and ixed ma hema ical s uc u e, an iso opy o iso opic li ing is any li ing o i , which gi es a new ma hema ical s uc u e e i ying he same basic axioms (o p ope ies) as he i s . This new s uc u e is called iso opic s uc u e o isos uc u e (see [2]). In 1978 (see [2]), San illi p oposes a possible model o iso opy, called San illi’s iso opy, which allows o cons uc he named ma hema ical isos uc u e, based on an isouni I. This isouni can be ob ained s a ing om he ollowing de ini ion: Le Ebe any ma hema ical s uc u e, de ined on a se o elemen s C. Le V⊇Cbe a se wi h an inne law ∗and an uni elemen I. Such a se Vis said o be he gene al se o he iso opy. Le b I∈Vbe such ha i s in e se T=b I−I, wi h espec o he law ∗, exis s. Iwill be called iso opic uni o isouni and i will be he basic uni in he li ing o he s uc u e E.Twill be he iso opic elemen . Finally, b Iand ∗a e he iso opy elemen s. Then, San illi p oposes o each an isos uc u e b Es a ing om he s uc u e E, by conside ing he ollowing cons uc ion le els: a) Con en ional le el: (see [2]) I is he ini ial ma hema ical s uc u e, o med by a se o elemen s and he laws de ined among hem. In his le el appea he usual ma hema ical s uc u es wi h espec o usual uni s: E=E(a, +,×, ...). b) Gene al le el: I is he gene al se V, in which a e, pa icula ly, he iso opy elemen s used in 2 Isog oups and isosubg oups he isop oduc cons uc ion model, ha is, V=V(α, ∗, ?, ...). I is impo an o no e ha E∗= E(a, ∗, ?, ...)( he es ic ion o V o E) mus e i y he same axioms as he ini ial s uc u e E. c) Iso opic le el: (see [2]) I is he ma hema ical isos uc u e ob ained when li ing, ha is b E= b E(ba, b +,b ×, ...). I is o med by an iso opic se and he isolaws on i . Elemen s o such se , which a e usually deno ed by using a ha , a e gi en wi h espec o he isouni o b E. So, ixed and gi en he isos uc u e (b E, b ×), wi h isouni b I, whe e Iis he uni o Ewi h espec o ∗, hese elemen s a e ba=bab ×b I, whe e San illi de ines he law b ×as bab ×b b=d a∗b. See hen ha , bab ×b I=d a∗I=ba=b Ib ×ba, which implies ha b Iis he uni elemen o b Ewi h espec o b ×. I is immedia e o check ha he mapping I:E→b E:a→bais a bijec ion, because i is on o by cons uc ion and i is also injec i e, due o ba6=b b, o all a, b ∈Esuch ha a6=b. Indeed, in b E, ba=bab ×b I6=b bb ×b I=b bwi h espec o he isouni b Io b E; in he same way as a=a×e6=b×e=b in E, whe e eis he uni elemen o Ewi h espec o ×. d) Le el o p ojec ion: (see [6]) I appea s when we conside he ma hema ical isos uc u e b E e e ed o he iso opy elemen s used in i s cons uc ion. I s elemen s a e deno ed by a line supe posed o he ha bo elemen s o b E, ha is,b. In his way, i we use he iso opy elemen s ∗(wi h uni I) and b I o cons uc b E, hen we ob ain a s uc u e b Ein he le el o p ojec ion, whose elemen s a e e e ed o he uni I:ba=a∗b I= (a∗b I)∗I. The mapping π:b E→b E:ba→π(ba) = bais named p ojec ion. In gene al, we say ha an elemen o b Eis p ojec ed on i s co esponding associa ed elemen belonging o b E. No e ha , by cons uc ion, he mapping πis on o. In a i s s age, b Eis only do ed wi h laws when πis linea wi h espec o he isolaws associa ed wi h b E. So, ixed an isos uc u e (b E, b ×), i πis linea wi h espec o b ×, he law b ×is de ined on b Eby bab ×b b=bab ×b b. In such a case, π: ( b E, b ×)7→ (b E, b ×)is an on o mo phism . The e o e, his le el o p ojec ion is he mos impo an in p ac ice, because i allows o ob ain some ma hema ical models which would be no possible unde usual uni s. The e exis s s ill ano he le el which joins bo h con en ional and iso opic le els. I is he axioma ic le el ([2]), which iden i ies e e y ma hema ical s uc u e e i ying he same axioms. So, in a schema ic way, as he di e en cons uc ion le els appea ing in an iso opic li ing, as he ela ions among hem can be obse ed in he ollowing diag am: Con en ional le el Gene al le el (V, ∗, ?, ...) −−−−−−−−−−−−−−−−→ (E, +,×, ...) (E, ∗, ?, ...) ↓]↓ Le el o p ojec ion P ojec ion ←−−−−−−−−−−−−−−− Iso opic le el (b E, b +,b ×, ...) ( b E, b +,b ×, ...) Finally, we will say ha an iso opic li ing o he s uc u e Eis injec i e i X=Y, o all X, Y ∈E such ha b X=b Y. I is equi alen , by cons uc ion, o say ha he p ojec ion π:b E→b E:ba→π(ba) = bais 3 R. M. Falc´ on and J. N´ u˜ nez an injec i e mapping. The e o e, as a consequence, i he iso opic li ing o Eis injec i e, hen π:b E7→ b E will be an isomo phism. 3.Isog oups S a ing om he de ini ion o isog oup (see [2]), we gi e in his sec ion some examples and p ope ies o hem. De ini ion 1 Le (G, ◦)be a g oup, wi h an associa i e inne law ◦, and an uni elemen e. An isog oup b Gis an iso opy o G, now equipped wi h a new inne associa i e law, b ◦and an uni elemen b I, such ha he pai (b G,b ◦) e i ies he axioms o a g oup. I , besides, bαb ◦b β=b βb ◦bαis e i ied o all bα, b β∈b G, hen we say ha b Gis an isoabelian isog oup o isocommu a i e isog oup. See ha his de ini ion o isog oup is qui e gene al. So, he uni elemen wi h espec o b ◦, which we call isouni , is no , in gene al, he men ioned isouni in a San illi’s iso opy. Howe e , when we do ha cons uc ion, we look a e o make i in his way so ha hese wo elemen s we e he same. Mo eo e , he ac o w i ing b Iin he place o be, is no casual. I we ollow he no a ion used jus he e, beis he iso opic li ing o e, bu , in gene al, beis no he uni elemen o b ◦. I implies ha no ions, p ope ies and heo ems s udied in he ini ial s uc u e canno be applied in he new s uc u e. Le see i in he case ha we ha e go a San illi’s iso opy, using a ixed isouni and a ∗law. To do i , when we ha e he ini ial g oup (G, ◦), we conside he isouni b I(which does no belong o Gin gene al), and we de ine he law ∗which we wan o wo k wi h. I is al eady known he model ha we use o cons uc he iso opic se b G, by using he isouni b Iand he law ∗(which is applied o any s uc u e). So, b G={bα=α∗b I=αb I|α∈G}. Now, le see how o li he associa ed law ◦. In he iso opic le el, we de ine he new law as ollows: bαb ◦b β=[ α∗β,∀bα, b β∈b G. The e o e, in he le el o p ojec ion, we de ine he new law as ollows: bαb ◦b β= (α∗β)∗b I. The new law is called isop oduc . We can show nex ha i we impose ha (G, ∗)is an associa i e g oup wi h I∈Gas he uni elemen wi h espec o ∗, hen (b G,b ◦)is an isog oup. To do i , we will see ha b ◦is an inne law which e i ies he axioms o a g oup. Indeed, ∀bα, b β, bγ∈b G, we ha e: a) b ◦is an inne law o b G, since bαb ◦b β=[ α∗β∈b G, because α∗β∈G, due o ∗is an inne law o Gby hypo hesis ( emembe ha (G, ∗)mus be a g oup). b) (bαb ◦b β)b ◦bγ=[ α∗βb ◦bγ= (α∗β)∗γ= α∗(β∗γ) = bαb ◦[ β∗γ=bαb ◦(b βb ◦bγ). (See ha ∗is associa i e is e y impo an so ha b ◦is associa i e). c) b I∈b G, since I∈G. (See ha b I=I∗b I). Mo eo e , b Iis he isouni ha we sea ch, because bαb ◦b I=[ α∗I=bα=b Ib ◦bα. d) Le bα∈b Gbe. I will be α∈G. So, as (G, ∗)is a g oup wi h uni elemen I, he e exis s α−I∈G, such ha α∗α−I=α−I∗α=I. Then, we mus only ake he elemen d α−I, as he isoin e se o bαwi h espec o b ◦,because hen we ha e ha bαb ◦d α−I= α∗α−I=b I=d α−Ib ◦bα. e) Finally, i ∗is commu a i e, (b G,b ◦)will be also commu a i e, because bαb ◦b β=[ α∗β=[ β∗α=b βb ◦bα. 4 Isog oups and isosubg oups So, we ha e p o ed he ollowing: Theo em 1 Le (G, ◦)be an associa i e g oup and le b Iand ∗be wo iso opy elemen s. I (G, ∗)is an associa i e g oup wi h uni elemen I∈G, hen he iso opic li ing (b G,b ◦)cons uc ed by he model o he isop oduc , has an isos uc u e o isog oup. Mo eo e , i (G, ∗)is commu a i e, hen (b G,b ◦)is a commu a i e isog oup. ¥ Now, we will see some examples o isog oups: Example 1 Le (R,+) be he g oup o he eal numbe s wi h he usual sum. A i ial iso opic li ing could be cons uc ed by using he isouni b I= 0 and he law ∗ ≡ +(no e ha (R,∗)=(R,+) is a g oup wi h uni elemen 0∈R); so we would ha e he pai (b R,b +),whe e b R={ba=a∗0 = a+ 0 = a|a∈R}=R.Mo eo e , as ∗ ≡ +, he isop oduc would be de ined as bab +b b=d a∗b=[ a+band bab +b b=[ a+b=a+b. So, we would ha e b +≡ ∗ ≡ +. So, he iso opy o (R,+), gi en by he isouni 0and he law ∗ ≡ +, is he same as he i ial iso opy, ha is, he iden i y. I p o es ha he cons uc ion which we a e using is igh , since i we do no change ei he he ini ial uni o he ini ial g oup law, hen his g oup emains in a ian when cons uc ing he iso opy. ¥ Example 2 Now, le conside he iso opy o he g oup (R∗,×)(R∗is he eal numbe s se minus he ze o), ob ained by using he isouni b I=iand he law ∗ ≡ • ( ha is, by he usual complex law). So, he iso opic se is c R∗=Im(C) {0}and he isop oduc will be de ined as bab ×b b=d a∗b=d a•b= [ a×b o all a, b ∈R.Then, bab ×b b=bab ×b b=[ a×b= (a×b)∗i= (a•b)•i o all a, b ∈R.¥ Mo eo e , le see ha he isog oups o he las wo examples a e isocommu a i e. I can be obse ed by using Theo em 1, because we ha e ha he ini ial g oups a e commu a i e. No e ha in bo h examples i has been used an isouni ac ing as a cons an . Howe e , examples in which he isouni used depends on ini ial coo dina es can be also shown: Example 3 Le conside (R,+), as in Example 1. We conside an iso opic li ing wi h iso opy elemen s ∗ ≡ +and b I=b I(x) = ½1, i x= 0 1 x2, i x6= 0 ¾. Then, b Iis posi i e de ined and non-singula and hus we ob ain he li ing a→ba=a∗b I=½0, i a= 0 1 a, i a6= 0 ¾. Finally, he isop oduc is de ined by bab +b b=[ a+b, whe e bab +b b=½0, i a+b= 0 1 a+b, i a+b6= 0 ¾in he p ojec- ion le el. In his way, he se {0}∪{1 a:a∈R∗}can be do ed o an isog oup s uc u e, by he law b +. ¥ To inish his sec ion we will p o e ha ixed and gi en an a bi a y isog oup, i can be conside ed as an iso opic li ing which ollows he iso opic cons uc ion model which we a e conside ing. Indeed, we ha e he ollowing: 5 R. M. Falc´ on and J. N´ u˜ nez P oposi ion 1 E e y iso opy I: (G, ◦)→(b G,b ◦)can be conside ed as an iso opic li ing which ollows he isop oduc cons uc ion model. PROOF. I is su icien o conside in he gene al le el he se (G, ∗), whe e ∗is de ined by a∗b=I−1(bab ◦b b) (i has sense, because he mapping I:E→b E:a→bais a bijec ion by cons uc ion, as we al eady obse ed). So, he isolaw b ◦can be de ined la e as bab ◦b b=I(a∗b) = d a∗b, and his is he way in which an isolaw is de ined acco ding o he isop oduc cons uc ion le el, as we al eady saw. In his way, we also ge , by linea i y, o do e he se (G, ∗)wi h a s uc u e o g oup. Mo eo e , he uni co esponding o ∗will be, by cons uc ion, he elemen o E, om which he isouni o he associa ed isolaw b ◦is iso opically li ed. ¥ 4.Isosubg oups In his sec ion we in oduce he de ini ion o isosubg oup. We gi e some examples and p ope ies o hem and we also deal wi h he di e ences be ween an isosubg oup and a subg oup o an isog oup. We s udy nex possible li ings o he subg oups, s a ing om he San illi’s cons uc ion model. To do his, we mus demand ha an isosubg oup is he iso opic li ing o a subg oup Ho a gi en and ixed g oup G. The di icul y appea s when we equi e ha e e y iso opic li ing o a gi en s uc u e is also a s uc u e o he same ype. So, e e y iso opy o Hshould ha e s uc u e o subg oup and hus, he iso opic li ings o Hcould no be independen o he li ing o G. The e o e, he de ini ion o isosubg oup would be as ollows: De ini ion 2 Le (G, ◦)be an associa i e g oup and (b G,b ◦)be an associa ed isog oup. Le Hbe a subg oup o G. We say ha b His an isosubg oup o b Gi , being an iso opy o H, he pai (b H, b ◦)is a subg oup o b G, ha is, i b H⊆b G,b ◦is an inne law in b Hand (b H, b ◦)has s uc u e o g oup. Le apply now his de ini ion o he San illi’s cons uc ion model, by an isouni and a law ∗. Suppose ha we ha e he associa i e g oup (G, ◦)and he isog oup (b G,b ◦), ob ained by bo h, an isouni b Iand a law ∗p e iously ixed. Le Hbe a subg oup o G. Since we demand ha in he u u e isosubg oup b H, he associa ed law is b ◦, i we go on abou he gi en cons uc ion o he isop oduc , hen he law and he isouni (bo h which will be he iso opy elemen s), mus be, espec i ely, ∗and b I, because i no , we would no ob ain he same law b ◦in gene al. See i wi h an example: Example 4 Le (Z,+) be he g oup o in ege s, wi h he usual sum. We ake, unde usual no a ions, ∗ ≡ +,b I= 2.As (Z,∗) = (Z,+) is a g oup wi h uni elemen 0∈Z,we can use he iso opy o elemen s ∗and b I. Then, b Z={ba=a∗2 = a+ 2 |a∈Z}=Zand he isop oduc is de ined as bab +b b=[ a+b, being bab +b b=[ a+b= (a+b)∗2 = a+b+2, o all a, b ∈Z.In his way, we ha e ob ained he isog oup (b Z,b +), which comes om he addi i e g oup (Z,+). Le now conside he subg oup (P,+) o e en in ege s and 0. I we cons uc he iso opy ela ed o he same elemen s as abo e (which is always possible, due o (P,∗) = (P,+) is a g oup wi h uni elemen 0∈P), we ob ain i s ly he iso opic se b P, being b P={bm=m∗2 = m+ 2 |m∈P}=P,and we would ge a e he same isop oduc b +. Le now p o e ha (b P,b +) is a isosubg oup o (b Z,b +), aking in o conside a ion ha b Pis an iso opy o P⊆Z. To see i , we obse e 6 Isog oups and isosubg oups a) b P⊆b Z,since P⊆Z. b) bmb +bn= m+n∈b P, o all m, n ∈P. So, b +is an inne law on b P. c) b Psa is ies he g oup condi ions, because associa i i y is om (b G,b ◦), he isouni b I= 2 = b 0belongs o b Pand bm−b I=d −m∈b P,∀m∈P,since bmb +d −m= m+ (−m) = b 0 = b I=d −mb +bm. So, (b P,b +) is an isosubg oup o (b Z,b +).¥ No e ha in his example, we can a oid some s eps when cons uc ing b H, once i is p o ed ha we can make he iso opy co esponding o elemen s used o cons uc b G. Indeed, emembe ha by using Theo em 1, condi ions o be sa is ied a e ha he pai (H, ∗)is a g oup, ha ing he uni elemen I, he same o (G, ∗). So, i we p oceed simila ly as we did when p o ing ha (b G,b ◦)had a g oup s uc u e by he isop oduc b ◦(ob ained s a ing o ∗), we ha e he condi ions needed: b +is an inne law on b P, he g oup axioms a e sa is ied by cons uc ion and inally, (b H, b ◦)is associa i e due o b ◦is associa i e on (b G,b ◦), o ∗ being i by hypo hesis. The e o e, i is p o ed he ollowing: Theo em 2 Le (G, ◦)be an associa i e g oup and (b G,b ◦)be he associa ed isog oup co esponding o he iso opy o elemen s b Iand ∗. Le Hbe a subg oup o G. I (H, ∗)has s uc u e o subg oup o (G, ∗), hen he iso opic li ing (b H, b ◦), co esponding o he iso opy o elemen s b Iand ∗, is a isosubg oup o b G. ¥ In ac , as he isog oup cons uc ion model al eady poin ed ou his condi ion, we can say ha i he iso opy co esponding o b Iand ∗can be made, hen (b H, b ◦)is an isosubg oup o b G. So, he las p oblem which could appea is ha such an iso opy could no be made o no e i ying some ini ial condi ions. We will see i in he ollowing example: Example 5 Le conside he g oup (Z/Z2,+) wi h he usual law +. Le w i e 1= 1 + Z/Z2and 2= 2 + Z/Z2. Conside now b I=1and he law ∗de ined by 1∗1=1=0∗0 1∗0=0∗1=0. I is easy o see ha ∗is associa i e, due o: (1∗1)∗1=1∗1=1∗(1∗1) (1∗1)∗0=1∗0=1∗(1∗0) (1∗0)∗0=0∗0=1=1∗1=1∗(0∗0) (0∗0)∗0=1∗0=0=0∗1=0∗(0∗0) whe eas o he possible cases a e also sa is ied by commu a i i y. The e o e, (Z/Z2,∗)has s uc u e o g oup, wi h uni elemen I=b I=1∈Z/Z2. Mo eo e , i we make now he iso opy co esponding o b Iand ∗, we ob ain ha he iso opic se is Z/Z2, being Z/Z2={b 0=0,b 1=1}=Z/Z2. 7 R. M. Falc´ on and J. N´ u˜ nez Besides, he co esponding isop oduc b +will be gi en by b 0b +b 0=[ 0∗0=b 1 b 0b +b 1=[ 0∗1=b 0=b 1b +b 0 b 1b +b 1=[ 1∗1=b 1 So, we ha e: 0b +0=b 0b +b 0=b 1=1 0b +1=b 0b +b 1=b 0=0=1b +0 1b +1=b 1b +b 1=b 1=1 Thus, b +≡ ∗ and so, ( Z/Z2,b +) is a new isog oup, being ( Z/Z2,b +) = (Z/Z2,∗). Le conside sepa a ely he subg oup ({0},+) o (Z/Z2,+). We obse e ha ({0},∗)does no ha e a s uc u e o g oup, because ∗is no an inne law on {0},due o 0∗0=1/∈ {0}.So, condi ions o Theo em 2 o ob ain an isosubg oup by making he iso opy o elemen b I=1and ∗, a e no sa is ied. We would ha e, in ac , ha d {0}={b 0}, whe e b 0b +b 0=b 1/∈d {0}.¥ We a e going o ask ou sel es a new ques ion. Suppose ha we ha e an associa i e g oup (G, ◦), wi h uni elemen I, and le (b G,b ◦)be he isog oup associa ed o he iso opy o elemen s b Iand ∗. We know ha e e y isosubg oup o b Ghas s uc u e o subg oup. We also know some examples in which subg oups o Gdo no gi e ise o isosubg oup o b G, by using he same b Iand ∗like iso opy elemen s. Then, we can inally ask i e e y subg oup o b Ghas a s uc u e o isosubg oup o (b G,b ◦), ha is, i i comes om he iso opic li ing o a subg oup o G. O cou se, we ha e al eady no ed ha i we ixe a subg oup (b H, b ◦)o (b G,b ◦), hen he law b ◦has o be he same in bo h pai s, so he co esponding elemen s o iso opic ha e o coincide. Tha is, i b His an isosubg oup, hen i ha e o come om an iso opy ha ing he same elemen s as he ones used in he cons uc ion o b G. So, he only possible subse o Gwhich would gi e ise o he possible isosubg oup would be H={a∈G:ba∈b H} ⊆ G. Howe e , he pai (H, ◦)is no a subg oup o (G, ◦)in gene al, as we can check in he ollowing: Example 6 Le conside bo h he g oup (Z/Z2,+) and he isog oup ( Z/Z2,b +) men ioned in he las example. We ha e he pai s ({0},+) and ({b 1},b +) espec i ely, as he only p ope subg oups o bo h. As we ha e jus seen, i we ake he subg oup b H= ({b 1},b +) o ( Z/Z2,b +), he only possible subse o Z/Z2 om which we could gi e a s uc u e o isog oup o b Hwould be H={1},because 1∗1=1,whe e b I=1is he isouni used in ha example o cons uc he iso opy. Howe e , ({1},+) is no a subg oup o (Z/Z2,+), because, o ins ance, +is no an inne law on {1}, due o 1+1=0.¥ So, wi h his example, he las ques ion abo e men ioned is answe ed in he nega i e. In his way, he link be ween g oups and isog oups is inally sol ed. 8 Isog oups and isosubg oups 5.Iso ep esen a ion o ini e isog oups In his sec ion we y o ha e some ela ions be ween San illi’s iso heo y and he s anda d heo y o ep e- sen a ion o g oups. Le us ecall ha a ep esen a ion o a ini e g oup Gis a pai (V, ρ), whe e Vis a ec o k-space and ρ is a g oup homomo phism ρ:G→Gl(V). To gene alize his concep on bo h iso opic and p ojec ion le els we i s ly gi e he de ini ion o isoho- momo phism o isog oups and secondly, o iso ep esen a ion o ini e isog oups. De ini ion 3 Le (G, ◦)and (G0,•)be wo g oups and le (b G,b ◦)and (c G0,b •)be associa ed isog oups, espec i ely. An isohomomo phism de ined be ween b Gand c G0is he iso opic li ing o any mapping ρ: G→G0, ha is, bρ:b G→c G0:bg→bρ(bg) = d ρ(g), e i ying bρ(bgb ◦b h) = bρ(bg)b •bρ(b h), o all Lbg,b h∈b G. No e ha by demanding he compa ibili y o he li ing used, we ob ain ha ρis a g oups homomo phism wi h espec o he laws o Gand G0. To see i we i s ly in oduce he ollowing esul , which is easy o p o e: P oposi ion 2 Le (G, ◦)be a g oup and (b G,b ◦)an associa ed isog oup. I b Ghas been ob ained s a ing om an iso opy compa ible wi h espec o ◦( ha is, bgb ◦b h=[ g◦h, o all g, h ∈G), hen Gand b Ga e isomo phic g oups. ¤ Mo eo e , i we conside he isop oduc cons uc ion model (which is always possible), we deduce he ollowing: Co olla y 1 Unde he hypo hesis o P oposi ion 2, i he iso opy used ollows he isop oduc cons uc ion model, hen we ha e, in he gene al le el o he g oup (G, ∗), ha (G, ∗)≡(G, ◦). PROOF. I is immedia e by cons uc ion, because ixed and gi en a, b ∈G, we ha e d a∗b=bab ◦b b=d a◦b. So, a∗b=a◦b.¥ I is now possible o p o e he ollowing: P oposi ion 3 Unde condi ions o De ini ion 3, i he iso opy used o cons uc b Gand c G0is compa ible wi h espec o ◦and •, espec i ely, hen bρis an isohomomo phism be ween b Gand c G0i and only i ρis an homomo phism be ween Gand G0. PROOF. a) ⇒ Le us suppose ha bρ:b G7→ c G0is an isohomomo phism. Then, ixed g, h ∈G, we ha e ρ(g◦h) = bρ([ g◦h) = bρ(bgb ◦b h) = bρ(bg)b •bρ(b h) = ρ(g)•ρ(h), and hus, ρ(g◦h) = ρ(g)•ρ(h), which implies ha ρis an homomo phism, due o gand ha e a bi a y in G. b) ⇐ Le us suppose ha ρ:G7→ G0is an homomo phism. Fixed gand hin G, we ha e ha bρ(bgb ◦b h) = bρ([ g◦h) = ρ(g◦h) = ρ(g)•ρ(h) = bρ(bg)b •bρ(b h). So, bρis an isohomomo phism. ¥ 9