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