Isogroups and isosubgroups

RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL.97 (1), 2003, pp. 1–12 ´ Algebra / Algebra Isogroups and isosubgroups Ra´ul M. Falc´ on and Juan N´u˜ nez Abstract. The main goal of this paper is to give a mathematical foundation, serious and consistent, to some parts of Santilli’s isotheory. We study the isotopic liftings of groups and subgroups and we also deal with the differences between an isosubgroup and a subgroup of an isogroup. Finally, some links between this isotheory and the standard groups theory, referred to representation and equivalence relations among groups are shown. Isogrupos e isosubgrupos Resumen. El principal objetivo de este art´ ıculo es proporcionar un fundamento matem´ atico, consistente y riguroso, a determinadas partes de la isoteor´ ıa de Santilli. En ´ el se realiza el levantamiento isot´ opico de los grupos y subgrupos, estudi´ andose asimismo la diferencia entre un isosubgrupo y un subgrupo de un isogrupo. Se muestran finalmente algunas relaciones entre esta isoteor´ ıa y la teor´ ıa standard de grupos, referentes a los temas de representaci´ on de grupos y de relaciones de equivalencia entre grupos. 1.Introduction In 1978, the Italian-American theoretical physicist and mathematic Ruggero Maria Santilli proposes a generalization of conventional Lie’s theory by using the concept of isotopy (in the Greek sense of being ”axiom-preserving”, also called isotopic lifting), which implies the origin of the actually known like Santilli’s isotheory (see [2]). To do this, he extends the basic unit I=(+1,diag(+1, ..., +1), ...) of the initial structure to a generalized unit b I=b I(x, • x,•• x, ..., µ, τ, ...), called isounit, which depends on the coordinate x and their derivatives, on the density µ, on the temperature τand, in general, on any magnitude of the physic environment of the system in which we are. By using it, Santilli does a step-by-step generalization of the more important mathematical structures, obtaining other new ones, characterized by the fact of having the same properties as the initial ones, while the new units satisfy more general conditions than the verified by the initial ones. Santilli gives the name of mathematical isostructures to these new structures. In this way, he studies isogroups,isorings,isofields,isovectorspaces and isoalgebras (see [3], [4], [5] and [11], for instance). It allowed him to get in a fast way some development of physical applications, principally in Quantum Mechanics and Dynamical Problems of particles and antiparticles. Santilli’s isotopies allow to map any given and fixed linear, local and canonical structure into its most general possible non-linear, non-local Presentado por Jes´ us Ildefonso D´ ıaz. Recibido: 6 de Febrero de 2002. Aceptado: 30 de Octubre de 2002. Palabras clave / Keywords: Isotheory, Lie-Santilli, isogroup, isosubgroup Mathematics Subject Classifications: 17B99. c °2003 Real Academia de Ciencias, Espa˜ na. 1 R. M. Falc´ on and J. N´ u˜ nez and non-canonical forms which are capable of reconstructing linearity, locality and canonicity in certain generalized isospaces and isofields within the fixed inertial coordinates of the observer. However, in the last years, Santilli has found some mathematical inconsistencies in his early formulation of the isotheory. Due to it, Santilli and other mathematicians have studied the isotopic liftings of Functional Analysis and Differential Calculus (see [1] and [6]). It has allowed to get some important applications in Physics (see [7], [8], [9] and [10], for instance). So, as Santilli’s isotheory needs even a consistent mathematical foundation, Santilli himself has proposed several subjects of research to the international mathematical-scientific community. One of them consists on proving the existence of isostructures corresponding to the lifting of structures already known, although they do not have a practical application in Physic. Santilli thinks that it would be good to convince the scientific about the relevance of his research, which would give bigger consistence and reliance to his isotheory. In this paper, we try to partially response to Santilli’s petition, by studying a possible lifting of the simplest algebraic structure: the group structure. However, as there is already some studies about it (see [11]), we complete them, give also some examples and show how the subgroups can be isotopically lifted by using the Santilli’s model to construct isoproducts. Getting this last question is the main goal of this paper. To do this, we previously give in Section 2 some basic definitions related to isotopic liftings. Section 3 is devoted to the study of isogroups. Next, in Section 4, we obtain Theorem 2, which assures the construction of an isosubgroup, giving some examples, too. We also distinguish between subgroups and isosubgroups of an isogroup. Last two sections are devoted to study some applications of Santilli’s isotheory: representation and equivalence relations among isogroups, respectively, and some links among these concepts and the standard groups theory. 2.Preliminaries Remember that for a given and fixed mathematical structure, an isotopy or isotopic lifting is any lifting of it, which gives a new mathematical structure verifying the same basic axioms (or properties) as the first. This new structure is called isotopic structure or isostructure (see [2]). In 1978 (see [2]), Santilli proposes a possible model of isotopy, called Santilli’s isotopy, which allows to construct the named mathematical isostructure, based on an isounit I. This isounit can be obtained starting from the following definition: Let Ebe any mathematical structure, defined on a set of elements C. Let V⊇Cbe a set with an inner law ∗and an unit element I. Such a set Vis said to be the general set of the isotopy. Let b I∈Vbe such that its inverse T=b I−I, with respect to the law ∗, exists. Iwill be called isotopic unit or isounit and it will be the basic unit in the lifting of the structure E.Twill be the isotopic element. Finally, b Iand ∗are the isotopy elements. Then, Santilli proposes to reach an isostructure b Estarting from the structure E, by considering the following construction levels: a) Conventional level: (see [2]) It is the initial mathematical structure, formed by a set of elements and the laws defined among them. In this level appear the usual mathematical structures with respect to usual units: E=E(a, +,×, ...). b) General level: It is the general set V, in which are, particularly, the isotopy elements used in 2 Isogroups and isosubgroups the isoproduct construction model, that is, V=V(α, ∗, ?, ...). It is important to note that E∗= E(a, ∗, ?, ...)(the restriction of Vto E) must verify the same axioms as the initial structure E. c) Isotopic level: (see [2]) It is the mathematical isostructure obtained when lifting, that is b E= b E(ba, b +,b ×, ...). It is formed by an isotopic set and the isolaws on it. Elements of such set, which are usually denoted by using a hat, are given with respect to the isounit of b E. So, fixed and given the isostructure (b E, b ×), with isounit b I, where Iis the unit of Ewith respect to ∗, these elements are ba=bab ×b I, where Santilli defines the law b ×as bab ×b b=d a∗b. See then that, bab ×b I=d a∗I=ba=b Ib ×ba, which implies that b Iis the unit element of b Ewith respect to b ×. It is immediate to check that the mapping I:E→b E:a→bais a bijection, because it is onto by construction and it is also injective, due to ba6=b b, for all a, b ∈Esuch that a6=b. Indeed, in b E, ba=bab ×b I6=b bb ×b I=b bwith respect to the isounit b Iof b E; in the same way as a=a×e6=b×e=b in E, where eis the unit element of Ewith respect to ×. d) Level of projection: (see [6]) It appears when we consider the mathematical isostructure b Ereferred to the isotopy elements used in its construction. Its elements are denoted by a line superposed to the hat bof elements of b E, that is,b. In this way, if we use the isotopy elements ∗(with unit I) and b Ito construct b E, then we obtain a structure b Ein the level of projection, whose elements are referred to the unit I:ba=a∗b I= (a∗b I)∗I. The mapping π:b E→b E:ba→π(ba) = bais named projection. In general, we say that an element of b Eis projected on its corresponding associated element belonging to b E. Note that, by construction, the mapping πis onto. In a first stage, b Eis only doted with laws when πis linear with respect to the isolaws associated with b E. So, fixed an isostructure (b E, b ×), if πis linear with respect to b ×, the law b ×is defined on b Eby bab ×b b=bab ×b b. In such a case, π: ( b E, b ×)7→ (b E, b ×)is an onto morphism . Therefore, this level of projection is the most important in practice, because it allows to obtain some mathematical models which would be no possible under usual units. There exists still another level which joins both conventional and isotopic levels. It is the axiomatic level ([2]), which identifies every mathematical structure verifying the same axioms. So, in a schematic way, as the different construction levels appearing in an isotopic lifting, as the relations among them can be observed in the following diagram: Conventional level General level (V, ∗, ?, ...) −−−−−−−−−−−−−−−−→ (E, +,×, ...) (E, ∗, ?, ...) ↓]↓ Level of projection Projection ←−−−−−−−−−−−−−−− Isotopic level (b E, b +,b ×, ...) ( b E, b +,b ×, ...) Finally, we will say that an isotopic lifting of the structure Eis injective if X=Y, for all X, Y ∈E such that b X=b Y. It is equivalent, by construction, to say that the projection π:b E→b E:ba→π(ba) = bais 3 R. M. Falc´ on and J. N´ u˜ nez an injective mapping. Therefore, as a consequence, if the isotopic lifting of Eis injective, then π:b E7→ b E will be an isomorphism. 3.Isogroups Starting from the definition of isogroup (see [2]), we give in this section some examples and properties of them. Definition 1 Let (G, ◦)be a group, with an associative inner law ◦, and an unit element e. An isogroup b Gis an isotopy of G, now equipped with a new inner associative law, b ◦and an unit element b I, such that the pair (b G,b ◦)verifies the axioms of a group. If, besides, bαb ◦b β=b βb ◦bαis verified for all bα, b β∈b G, then we say that b Gis an isoabelian isogroup or isocommutative isogroup. See that this definition of isogroup is quite general. So, the unit element with respect to b ◦, which we call isounit, is not, in general, the mentioned isounit in a Santilli’s isotopy. However, when we do that construction, we look after to make it in this way so that these two elements were the same. Moreover, the fact of writing b Iin the place of be, is not casual. If we follow the notation used just here, beis the isotopic lifting of e, but, in general, beis not the unit element of b ◦. It implies that notions, properties and theorems studied in the initial structure cannot be applied in the new structure. Let see it in the case that we have got a Santilli’s isotopy, using a fixed isounit and a ∗law. To do it, when we have the initial group (G, ◦), we consider the isounit b I(which does not belong to Gin general), and we define the law ∗which we want to work with. It is already known the model that we use to construct the isotopic set b G, by using the isounit b Iand the law ∗(which is applied for any structure). So, b G={bα=α∗b I=αb I|α∈G}. Now, let see how to lift the associated law ◦. In the isotopic level, we define the new law as follows: bαb ◦b β=[ α∗β,∀bα, b β∈b G. Therefore, in the level of projection, we define the new law as follows: bαb ◦b β= (α∗β)∗b I. The new law is called isoproduct. We can show next that if we impose that (G, ∗)is an associative group with I∈Gas the unit element with respect to ∗, then (b G,b ◦)is an isogroup. To do it, we will see that b ◦is an inner law which verifies the axioms of a group. Indeed, ∀bα, b β, bγ∈b G, we have: a) b ◦is an inner law for b G, since bαb ◦b β=[ α∗β∈b G, because α∗β∈G, due to ∗is an inner law for Gby hypothesis (remember that (G, ∗)must be a group). b) (bαb ◦b β)b ◦bγ=[ α∗βb ◦bγ=\ (α∗β)∗γ=\ α∗(β∗γ) = bαb ◦[ β∗γ=bαb ◦(b βb ◦bγ). (See that ∗is associative is very important so that b ◦is associative). c) b I∈b G, since I∈G. (See that b I=I∗b I). Moreover, b Iis the isounit that we search, because bαb ◦b I=[ α∗I=bα=b Ib ◦bα. d) Let bα∈b Gbe. It will be α∈G. So, as (G, ∗)is a group with unit element I, there exists α−I∈G, such that α∗α−I=α−I∗α=I. Then, we must only take the element d α−I, as the isoinverse of bαwith respect to b ◦,because then we have that bαb ◦d α−I=\ α∗α−I=b I=d α−Ib ◦bα. e) Finally, if ∗is commutative, (b G,b ◦)will be also commutative, because bαb ◦b β=[ α∗β=[ β∗α=b βb ◦bα. 4 Isogroups and isosubgroups So, we have proved the following: Theorem 1 Let (G, ◦)be an associative group and let b Iand ∗be two isotopy elements. If (G, ∗)is an associative group with unit element I∈G, then the isotopic lifting (b G,b ◦)constructed by the model of the isoproduct, has an isostructure of isogroup. Moreover, if (G, ∗)is commutative, then (b G,b ◦)is a commutative isogroup. ¥ Now, we will see some examples of isogroups: Example 1 Let (R,+) be the group of the real numbers with the usual sum. A trivial isotopic lifting could be constructed by using the isounit b I= 0 and the law ∗ ≡ +(note that (R,∗)=(R,+) is a group with unit element 0∈R); so we would have the pair (b R,b +),where b R={ba=a∗0 = a+ 0 = a|a∈R}=R.Moreover, as ∗ ≡ +, the isoproduct would be defined as bab +b b=d a∗b=[ a+band bab +b b=[ a+b=a+b. So, we would have b +≡ ∗ ≡ +. So, the isotopy of (R,+), given by the isounit 0and the law ∗ ≡ +, is the same as the trivial isotopy, that is, the identity. It proves that the construction which we are using is right, since if we do not change either the initial unit or the initial group law, then this group remains invariant when constructing the isotopy. ¥ Example 2 Now, let consider the isotopy of the group (R∗,×)(R∗is the real numbers set minus the zero), obtained by using the isounit b I=iand the law ∗ ≡ • (that is, by the usual complex law). So, the isotopic set is c R∗=Im(C)\ {0}and the isoproduct will be defined as bab ×b b=d a∗b=d a•b= [ a×bfor all a, b ∈R.Then, bab ×b b=bab ×b b=[ a×b= (a×b)∗i= (a•b)•ifor all a, b ∈R.¥ Moreover, let see that the isogroups of the last two examples are isocommutative. It can be observed by using Theorem 1, because we have that the initial groups are commutative. Note that in both examples it has been used an isounit acting as a constant. However, examples in which the isounit used depends on initial coordinates can be also shown: Example 3 Let consider (R,+), as in Example 1. We consider an isotopic lifting with isotopy elements ∗ ≡ +and b I=b I(x) = ½1, if x= 0 1 x2, if x6= 0 ¾. Then, b Iis positive defined and non-singular and thus we obtain the lifting a→ba=a∗b I=½0, if a= 0 1 a, if a6= 0 ¾. Finally, the isoproduct is defined by bab +b b=[ a+b, where bab +b b=½0, if a+b= 0 1 a+b, if a+b6= 0 ¾in the projection level. In this way, the set {0}∪{1 a:a∈R∗}can be doted of an isogroup structure, by the law b +. ¥ To finish this section we will prove that fixed and given an arbitrary isogroup, it can be considered as an isotopic lifting which follows the isotopic construction model which we are considering. Indeed, we have the following: 5 R. M. Falc´ on and J. N´ u˜ nez Proposition 1 Every isotopy I: (G, ◦)→(b G,b ◦)can be considered as an isotopic lifting which follows the isoproduct construction model. PROOF. It is sufficient to consider in the general level the set (G, ∗), where ∗is defined by a∗b=I−1(bab ◦b b) (it has sense, because the mapping I:E→b E:a→bais a bijection by construction, as we already observed). So, the isolaw b ◦can be defined later as bab ◦b b=I(a∗b) = d a∗b, and this is the way in which an isolaw is defined according to the isoproduct construction level, as we already saw. In this way, we also get, by linearity, to dote the set (G, ∗)with a structure of group. Moreover, the unit corresponding to ∗will be, by construction, the element of E, from which the isounit of the associated isolaw b ◦is isotopically lifted. ¥ 4.Isosubgroups In this section we introduce the definition of isosubgroup. We give some examples and properties of them and we also deal with the differences between an isosubgroup and a subgroup of an isogroup. We study next possible liftings of the subgroups, starting from the Santilli’s construction model. To do this, we must demand that an isosubgroup is the isotopic lifting of a subgroup Hof a given and fixed group G. The difficulty appears when we require that every isotopic lifting of a given structure is also a structure of the same type. So, every isotopy of Hshould have structure of subgroup and thus, the isotopic liftings of Hcould not be independent of the lifting of G. Therefore, the definition of isosubgroup would be as follows: Definition 2 Let (G, ◦)be an associative group and (b G,b ◦)be an associated isogroup. Let Hbe a subgroup of G. We say that b His an isosubgroup of b Gif, being an isotopy of H, the pair (b H, b ◦)is a subgroup of b G, that is, if b H⊆b G,b ◦is an inner law in b Hand (b H, b ◦)has structure of group. Let apply now this definition to the Santilli’s construction model, by an isounit and a law ∗. Suppose that we have the associative group (G, ◦)and the isogroup (b G,b ◦), obtained by both, an isounit b Iand a law ∗previously fixed. Let Hbe a subgroup of G. Since we demand that in the future isosubgroup b H, the associated law is b ◦, if we go on about the given construction of the isoproduct, then the law and the isounit (both which will be the isotopy elements), must be, respectively, ∗and b I, because if not, we would not obtain the same law b ◦in general. See it with an example: Example 4 Let (Z,+) be the group of integers, with the usual sum. We take, under usual notations, ∗ ≡ +,b I= 2.As (Z,∗) = (Z,+) is a group with unit element 0∈Z,we can use the isotopy of elements ∗and b I. Then, b Z={ba=a∗2 = a+ 2 |a∈Z}=Zand the isoproduct is defined as bab +b b=[ a+b, being bab +b b=[ a+b= (a+b)∗2 = a+b+2, for all a, b ∈Z.In this way, we have obtained the isogroup (b Z,b +), which comes from the additive group (Z,+). Let now consider the subgroup (P,+) of even integers and 0. If we construct the isotopy related to the same elements as above (which is always possible, due to (P,∗) = (P,+) is a group with unit element 0∈P), we obtain firstly the isotopic set b P, being b P={bm=m∗2 = m+ 2 |m∈P}=P,and we would get after the same isoproduct b +. Let now prove that (b P,b +) is a isosubgroup of (b Z,b +),taking into consideration that b Pis an isotopy of P⊆Z. To see it, we observe 6 Isogroups and isosubgroups a) b P⊆b Z,since P⊆Z. b) bmb +bn=\ m+n∈b P, for all m, n ∈P. So, b +is an inner law on b P. c) b Psatisfies the group conditions, because associativity is from (b G,b ◦), the isounit b I= 2 = b 0belongs to 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 isosubgroup of (b Z,b +).¥ Note that in this example, we can avoid some steps when constructing b H, once it is proved that we can make the isotopy corresponding to elements used to construct b G. Indeed, remember that by using Theorem 1, conditions to be satisfied are that the pair (H, ∗)is a group, having the unit element I, the same of (G, ∗). So, if we proceed similarly as we did when proving that (b G,b ◦)had a group structure by the isoproduct b ◦(obtained starting of ∗), we have the conditions needed: b +is an inner law on b P, the group axioms are satisfied by construction and finally, (b H, b ◦)is associative due to b ◦is associative on (b G,b ◦), for ∗ being it by hypothesis. Therefore, it is proved the following: Theorem 2 Let (G, ◦)be an associative group and (b G,b ◦)be the associated isogroup corresponding to the isotopy of elements b Iand ∗. Let Hbe a subgroup of G. If (H, ∗)has structure of subgroup of (G, ∗), then the isotopic lifting (b H, b ◦), corresponding to the isotopy of elements b Iand ∗, is a isosubgroup of b G. ¥ In fact, as the isogroup construction model already pointed out this condition, we can say that if the isotopy corresponding to b Iand ∗can be made, then (b H, b ◦)is an isosubgroup of b G. So, the last problem which could appear is that such an isotopy could not be made for not verifying some initial conditions. We will see it in the following example: Example 5 Let consider the group (Z/Z2,+) with the usual law +. Let write 1= 1 + Z/Z2and 2= 2 + Z/Z2. Consider now b I=1and the law ∗defined by 1∗1=1=0∗0 1∗0=0∗1=0. It is easy to see that ∗is associative, due to: (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) whereas other possible cases are also satisfied by commutativity. Therefore, (Z/Z2,∗)has structure of group, with unit element I=b I=1∈Z/Z2. Moreover, if we make now the isotopy corresponding to b Iand ∗, we obtain that the isotopic set 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, the corresponding isoproduct b +will be given 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 have: 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 isogroup, being (\ Z/Z2,b +) = (Z/Z2,∗). Let consider separately the subgroup ({0},+) of (Z/Z2,+). We observe that ({0},∗)does not have a structure of group, because ∗is not an inner law on {0},due to 0∗0=1/∈ {0}.So, conditions of Theorem 2 to obtain an isosubgroup by making the isotopy of element b I=1and ∗, are not satisfied. We would have, in fact, that d {0}={b 0}, where b 0b +b 0=b 1/∈d {0}.¥ We are going to ask ourselves a new question. Suppose that we have an associative group (G, ◦), with unit element I, and let (b G,b ◦)be the isogroup associated to the isotopy of elements b Iand ∗. We know that every isosubgroup of b Ghas structure of subgroup. We also know some examples in which subgroups of Gdo not give rise to isosubgroup of b G, by using the same b Iand ∗like isotopy elements. Then, we can finally ask if every subgroup of b Ghas a structure of isosubgroup of (b G,b ◦), that is, if it comes from the isotopic lifting of a subgroup of G. Of course, we have already noted that if we fixe a subgroup (b H, b ◦)of (b G,b ◦), then the law b ◦has to be the same in both pairs, so the corresponding elements of isotopic have to coincide. That is, if b His an isosubgroup, then it have to come from an isotopy having the same elements as the ones used in the construction of b G. So, the only possible subset of Gwhich would give rise to the possible isosubgroup would be H={a∈G:ba∈b H} ⊆ G. However, the pair (H, ◦)is not a subgroup of (G, ◦)in general, as we can check in the following: Example 6 Let consider both the group (Z/Z2,+) and the isogroup (\ Z/Z2,b +) mentioned in the last example. We have the pairs ({0},+) and ({b 1},b +) respectively, as the only proper subgroups of both. As we have just seen, if we take the subgroup b H= ({b 1},b +) of (\ Z/Z2,b +), the only possible subset of Z/Z2from which we could give a structure of isogroup to b Hwould be H={1},because 1∗1=1,where b I=1is the isounit used in that example to construct the isotopy. However, ({1},+) is not a subgroup of (Z/Z2,+), because, for instance, +is not an inner law on {1}, due to 1+1=0.¥ So, with this example, the last question above mentioned is answered in the negative. In this way, the link between groups and isogroups is finally solved. 8 Isogroups and isosubgroups 5.Isorepresentation of finite isogroups In this section we try to have some relations between Santilli’s isotheory and the standard theory of representation of groups. Let us recall that a representation of a finite group Gis a pair (V, ρ), where Vis a vector k-space and ρ is a group homomorphism ρ:G→Gl(V). To generalize this concept on both isotopic and projection levels we firstly give the definition of isohomomorphism of isogroups and secondly, of isorepresentation of finite isogroups. Definition 3 Let (G, ◦)and (G0,•)be two groups and let (b G,b ◦)and (c G0,b •)be associated isogroups, respectively. An isohomomorphism defined between b Gand c G0is the isotopic lifting of any mapping ρ: G→G0, that is, bρ:b G→c G0:bg→bρ(bg) = d ρ(g), verifying bρ(bgb ◦b h) = bρ(bg)b •bρ(b h), for all Lbg,b h∈b G. Note that by demanding the compatibility of the lifting used, we obtain that ρis a groups homomorphism with respect to the laws of Gand G0. To see it we firstly introduce the following result, which is easy to prove: Proposition 2 Let (G, ◦)be a group and (b G,b ◦)an associated isogroup. If b Ghas been obtained starting from an isotopy compatible with respect to ◦(that is, bgb ◦b h=[ g◦h, for all g, h ∈G), then Gand b Gare isomorphic groups. ¤ Moreover, if we consider the isoproduct construction model (which is always possible), we deduce the following: Corollary 1 Under the hypothesis of Proposition 2, if the isotopy used follows the isoproduct construction model, then we have, in the general level of the group (G, ∗), that (G, ∗)≡(G, ◦). PROOF. It is immediate by construction, because fixed and given a, b ∈G, we have d a∗b=bab ◦b b=d a◦b. So, a∗b=a◦b.¥ It is now possible to prove the following: Proposition 3 Under conditions of Definition 3, if the isotopy used to construct b Gand c G0is compatible with respect to ◦and •, respectively, then bρis an isohomomorphism between b Gand c G0if and only if ρis an homomorphism between Gand G0. PROOF. a) ⇒ Let us suppose that bρ:b G7→ c G0is an isohomomorphism. Then, fixed g, h ∈G, we have \ ρ(g◦h) = bρ([ g◦h) = bρ(bgb ◦b h) = bρ(bg)b •bρ(b h) = \ ρ(g)•ρ(h), and thus, ρ(g◦h) = ρ(g)•ρ(h), which implies that ρis an homomorphism, due to gand hare arbitrary in G. b) ⇐ Let us suppose that ρ:G7→ G0is an homomorphism. Fixed gand hin G, we have that bρ(bgb ◦b h) = bρ([ g◦h) = \ ρ(g◦h) = \ ρ(g)•ρ(h) = bρ(bg)b •bρ(b h). So, bρis an isohomomorphism. ¥ 9