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Algebraic Approximations to Partial Group Structures

Written By

Özen Özer

Reviewed: December 20th, 2021 Published: February 22nd, 2022

DOI: 10.5772/intechopen.102146

IntechOpen
Coding Theory - Recent Advances, New Perspectives and Applications Edited by Sudhakar Radhakrishnan

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Coding Theory - Recent Advances, New Perspectives and Applications [Working Title]

Dr. Sudhakar Radhakrishnan and Dr. Sudev Naduvath

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Abstract

In this work, we use ‘Partial Group’ notion and we do further investigations about partial groups. We define ‘Partial Normal Subgroup’ using partial conjugation criteria and we prove few results about partial normal subgroups analogous to normal groups. Also, we define congruence relation for partial groups and via this relation, we state ‘The Quotient of Partial Group or Factor Group’. We give isomorphism theorems for partial groups. Explicitly, this is an analogous concept to group theory and our main is where differences partial groups from groups.

Keywords

  • partial group
  • partial normal subgroup
  • partial quotient group
  • isomorphism theorems for partial groups

1. Introduction

It is defined that a group is a set equipped with an operation described on it such that it has some properties as associated elements, an identity element, and inverse elements. Another definition can also be given as algebraic that the group is the set of all the permutations for algebraic expression’s roots that displays the typical that the assembly of the permutations pertains to the set.

If questions are “how was group theory developed?, What is the importance of group theory in science or real life? investigated for the mathematical topic group theory, then we can understand easily why we work on the structures of the theory of the many types of groups.

As we know from the literature, some primary sources are determined in the development of group theory such as Algebra (Lagrange in the 17. century), Number Theory (Gauss in the 18. century) (Euler’s product formula, Combinatorics, Fermat’s Last Theorem, Class group, Regular primes, Burnside’s lemma), Geometry (Klein, 1874), and Analysis (Lie, Poincaré, Klein in the 18. Century). It seems that three main areas have been described as Number Theory and Algebra (Galois theory, equation with degree 5, Class field theory), Geometry (Torus, Elliptic curves, Toric varieties, Resolution of singularities), and analysis in mathematics. Topology (((co)homology groups, homotopy groups) and algebraic part of it (Eilenberg–MacLane spaces, Torsion subgroups, Topological spaces), the Theory of Manifolds (manifolds with a metric), Algebra, Dynamical systems, Engineering (to create digital holograms), Combinatorial Number Theory, Mathematical Logic, Geometry in Riemannian Space, and Lie Algebra also belongs these three subjects.

Group theory is used not just in mathematics but also in computer science, physics, chemistry, engineering, and other sciences. Especially symmetry has a big potential property in the group theory. That is why it is considered as representation theory in physics. For example; mathematical works on quantum mechanics were done by von Neumann, Molecular Orbital Theory. Also, the Standard Model of particle physics, the equations of motion, or the energy eigenfunctions use group theory for their orbitals, classify crystal structures, Raman and infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy or getting periodic tables-gauge theory, the Lorentz group-the Poincaré’s group in modern chemistry or physics. Also, group theory is defined as representation theory in physics. A lot of groups with prime caliber built-in cryptography for elliptic curve do a service for public-key cryptography and Diffie–Hellman key exchange takes advantage of cyclic groups (especially finite) too. Additionally, cryptographic protocols also consider infinite nonabelian groups.

We can state the applications of the group theory also in real life as follows:

  1. Shopping online (we use our credit card with encryption which is obtained by group structure in RSA algorithm)

  2. Music (Elementary group theory is used for the 12-periodicity in the circle of fifths in musical set theory. Transformational theory patterns musical transformations as if they are elements of a mathematical group, cyclic groups create octave and other notions, the musical actions of the dihedral groups.)

  3. Medical science (to find out breast cancer) and computer science (robotics computer vision and computer graphics) and material sciences.

  4. Machine learning, communication network, signal processing, etc.…

  5. Pipeline system, which is described as the Application - Business Object - Network Node Layer, is patterned and investigated by the theory of group. These systems are also related with vectors, matrices determined by group structures, and so on.

Thus, tools of the group theory are useful for working on applications in many different sciences and also real life as mentioned above.

Basic and simple examples can be given for usual groups as follows:

  • Vector spaces V+have group structures under the addition of vectors with some properties of the scalar multiplication *.

  • For pprimes, elements of ZpZphave algebraic structures under multiplication with unit 1.

  • Assume that Fbe a number field and mis a natural number. Then S=SLmFcan be determined to be the set of all regular m×mmatrices with logins in F. This is a usual group with abdescribed by the multiplication of matrices.

We can ask readers “whether or not these examples are partial group”?

Smis demonstrated by symmetric groups such that it includes m!permutations where mobjects are taken from a set A. As an illustration, we can give the symmetric group S3. Supposing that A=abcand S3contains following objects; identity element.

Identity element=abcabcand othersare;
abcbac,abcacb,abccba,abcbca,abacab

There are some properties in finite or infinite usual group theory. Some of them can be seen as follows:

  • Each of the elements in the finite group has finite order.

  • If His a finite group, then it is satisfied for each of the subgroups of H.

  • Assuming that X,Yare groups. Then, the product of them is defined by X×Y=abaXbYwith unit element eeXeY,where we write eX,eYare identity element and inverse elements are in the form of a1b1of in X,Y, respectively.

  • Suppose that Xbe a usual group and ain X. So, the cyclic group abecomes a subgroup of X. As an illustration, we can say that (Zm, +) is a cyclic group that satisfies 1=Zm.

  • IfXbe a usual group with order p(pis prime), Xis also a cyclic group.

  • Xis abelian usual group iff center of Xequals to X.

  • Let us consider Smand its two permutations. These are conjugate iff their cycle types are equal/same according to their ordering.

  • ….

If literature (briefly, references [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]) is investigated, then it is easily seen that partial groups are considered as topological structures more than algebraic structures. It is tried to prepare some new algebraic perspectives/approximations for the partial group. As we know, there are many algebraic infrastructures such as finite-infinite group, abelian-nonabelian group, quaternion group, symmetric group, cyclic group, simple group, free group, orbits and stabilizers of the group, Lie group, and various kinds of theorems such as Sylow theorems, Cauchy theorem, Lagrange theorem, Cayley theorem, Isomorphism theorems as well as actions of groups for usual group theory.

An effect algebra is introduced in the foundations of mechanics [1]. Furthermore, effect algebra subjects are fundamental in fuzzy probability theory [2, 3]. Also, partial group is defined by [4] and used for topological and homological investigations. A pregroup can be defined as following:

A pregroup, [5], of a set Pcontaining an element 1, each element pPhas a unique inverse p−1 and to each pair of elements p,tPthere is defined at most one product ptPso that;

  1. 1p= p1=pis always defined,

  2. p*p−1 =p−1 * p= 1 is always defined,

  3. If ptis defined then t−1p−1 is defined and equal to (pt)−1.

  4. If rpand ptare defined then either rptis defined if and only if rptis defined in which case two are equal.

  5. If qr,rpand ptare defined then either qrpis defined or rptis defined.

Every pregroup is a partial group, but the converse is not true in general. In that meaning a partial group definition can be stated as follows:

A set Pis a partial group in the meaning of ([6], Lemma 4.2.5) if each associated pair xyPXPthere is at most one product x.yso that:

  1. There is an element 1 Psatisfying x.1 = 1. x=xfor each xP.

  2. For each xPthere exists an element x−1 so that x.x−1=x−1.x=1

  3. If x.y=zis defined so is y−1.x−1 = z−1.

Inspiring by the groupoid and effect algebra [7] gave an alternative partial group definition as an algebraic style. They introduced partial subgroup, partial group homomorphism, etc. as an analog investigation for group theory. Moreover, readers can learn/consider a lot more structural results on the subject from others [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47].

In this work, some (remained ones can be considered from readers using our works) fundamental results are given for partial groups. Similarities and differences have been noticed between usual groups and partial groups. Several of them also are described in this work.

Also, we do further investigations for partial groups in algebraic style. We define partial normal subgroup and give isomorphism theorems for partial groups. This work is important because it has both topological and algebraic applications. It can be expanded to rings or other algebraic structures.

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2. Preliminary results

Recently, an algebraic structure named as partial group (also known as Clifford Semigroup is isomorphic to an explicit partial group of partial mappings and it is a semigroup with central idempotents) is investigated with new structures in the literature.

A partial group (Clifford semigroup) is a regular semigroup (it means if Mis a semigroup of group G then idempotent elements of Mexchange with H’s all elements). Another definition can be given for the partial group as “A regular semigroup with its central idempotents is named by Clifford semigroup.”

Additionally, several partial algebras such as partial monoid, partial ring, partial group ring, partial quasigroup etc.… have been worked. For example, Jordan Holder Theorem of composition series is known to hold in every abelian category. The classical theory of subnormal series, refinements, and composition series in groups is extended to the class of partial groups which is known to be precisely the classes of Clifford semigroups, or equivalently semilattices of groups. Also, relations among the language theory, words, partial groups, universal group, and homology theory have been considered with an arrow diagram of the partial group.

In this chapter, we first state some basic properties of partial groups which are mentioned in several references.

Definition 2.1[7] Suppose G* is a nonempty set: G* is called as a partial group if the following conditions hold for all x,y,and zG*:

(G1) If xy,xyz,yzandxyzare defined, then the equality xyz=xyzis valid.

(G2) For eachxG*, there exists an eG* such that xeand exare defined and the equality xe=ex=xis valid.

(G3) For each xG*, there exists an xG* such that xxandxxare defined and the equality xx=xx=eis valid.

The eG* satisfies (G2) is called the identity element of G* and the XG* satisfies (G3) is called the inverse of xand denoted by x−1.

Another way, we can give partial definition as follows:

Definition 2.2.Let Mbe a semigroup. It is called apartial group if the followings are held.

  1. Every kMhas a partial identity ek

  2. Every kMhas a partial inverse k1

  3. Mapping eM:MM,kekis a semigroup homomorphism

  4. Mapping β:MM,kkis a semigroup antihomomorphism.

Note:

  1. Let Mbe a semigroup. A partial identity of k, when exists, is unique and idempotent such that denoted by ek.

  2. Let Mbe a semigroup. A partial inverse of kϵM, when exists, is uniquely denoted by k1.

Definition 2.3.The regular element of the Msemigroup is defined if there exists s ∈Msuch that ysy = y. Each element ofMis regular element also Mis named by regular semigroup.

Definition 2.4.Msemigroup is called as completely regular semigroup for every element s Mysy = y and ys = sy are satisfied.

Note.Unions of groups give us completely regular semigroups which are named by Clifford semigroups.

Definition 2.5.Let Mbe a semigroup. Elements k and s of a semigroup Mare said to be inverse of each other if and only if sks = s and ksk = k.

Then following theorem can be given from the literature.

Theorem 2.1.The following results are equivalent to each other for a semigroup M:

  1. Mis a Clifford semigroup,

  2. There exists r ∈ S such that wrw = w and wr = rw for every w ∈M,

  3. Mis a semilattice of groups,

  4. Mis a completely regular inverse semigroup.

Proposition 2.1.Mis a completely regular semigroup iff there are ekand k1for every kϵM.

Proposition 2.2[7] Every group is a partial group and every partial group which is closed under its partial group operation is a group.

Proposition 2.3.Assuming that Mis a partial group. Then, the following are given:

  • Every idempotent element in partial group Mis its own partial identity and partial inverse,

  • LetkϵM.Then,ek1=ek=ek1is hold.

  • Suppose that kϵM.k11=kis satisfied.

Example 2.1[7] Following sets with the given operation, can be seen as an example to the partial groups:

  1. Let G={0,±1,…,±n} where nZ+ and + be known addition operation on Z. Then it is easily seen that Gis a partial group but is not a group.

  2. Let G=Z*{1n:nZ+}where Z*=Z0.So it is obvious that Gis a partial group but is not a group by the known multiplication on R.

  3. Let G=rrwhere rR+ and + be known addition operations on R. Then it is obvious Gis a partial group but is not a group.

Definition 2.6[7] Suppose G* be a partial group, mZ+, and aG*. If am is defined and mis the least integer such that am=e,the number mis called the order ofa. In this case, it is called that ahas a finite order element. If there does not exist an mZ+ such that am=e,(if only a0=e); then it is called that ahas infinite order. The order of a is denoted by a.

Example 2.2[7] G=11ii2ii2with the multiplication operation on Cis a partial group and i=4, 2i=.

Definition 2.7[7]. Suppose G*be a partial group. Z(G*)={xG*Ifaxandxaaredefined forallaA;ax=xa}is called the center of G*.

Lemma 2.1[7] A partial group is called centerlessif Z (G*) is trivial i.e., consists of only the identity element. If G* is commutative then G* = Z (G*).

Definition 2.8.Supposing that M=Gpbe a partial group and T=Hpbe a subset of M.Tis called by sub partial group of Mif Tis a sub semigroup of Mand ek,k1are in Tfor all kT.

Especially, Mand the set of idempotents elements of Mare sub partial groups of M.

Definition 2.9[7] Let G* be a partial group and H* be a nonempty subset of G*. If H* is a partial group with the operation in G* then H* is called a partial subgroup of G*.

Example 2.3[7] In Example 2.2 the set G*=0±1±nwhere nϵZ+ and + be known addition operation on Zis a partial group and let H*=0±1±kwhere 0knand kZ. Then H* is a partial subgroup of G*.

Lemma 2.2[7] Let G* be a partial group and H* be a nonempty subset of G*.H* is a partial subgroup of G* if and only if the following conditions hold:

  1. eH*;

  2. a−1H* for all aH*.

Moreover, LetG* be a partial group and let abe an element of G* such that the elements {ak for all kZ} are defined. Denote{ak;kZ}=<a>It is clear that the set <a>is a partial subgroup of G*. The partial subgroup <a>of G* is called the cyclic partial subgroup generated by a. If there exists an element ain G* such that <a>=G*, then G* is called a cyclic partial group.

Example 2.4[7] Let G=eabcS=eabcdand “.” be a partially defined operation on G as in Table 1.

.eabc
eeabc
aabce
bbcea
ccedb

Table 1.

(G) is a partial group even it has not a group structure.

Remark.Note that c.bis undefined. Then Gis not a group but it is a partial group. Additionally, in this partial group, <a>=G, Gis the cyclic partial group. But all partial subgroups of a cyclic partial group can not be cyclic. For instance, the partial subgroup H=eacis not cyclic. But in group theory, if a group is cyclic, all subgroups of it are also cyclic. The partial groups are different from groups in that meaning.

Definition 2.10.Assume that M=Gpbe apartial group and kM.Then, we define Mk=sM:ek=es.

Theorem 2.2.Suppose that Mis apartial group and kM.Then, Mkis a maximal subgroup Mof which has identity ekand M=Mk:kM.

Definition 1.11[7] Let Mand Nbe partial groups. A function σ:MNis called a partial group homomorphism if for all a,bMsuch that abis defined in M, σaσ(b) is defined in Nand

σab=σaσb.

If σis injective as a map of sets, σis said to be a monomorphism. If σis surjective, σis called an epimorphism.

Definition 2.12.For a partial group homomorphism σ:MNit is defined kerσ=kM:σk=eσkand Imσ=σk:kM. Also, σ:MNis named isomorphism if it is bijective.

As a consequence of the definition the following lemmas can be given:

Definition 2.13.Suppose that σ:MNbe a partial group homomorphism. Then, we define ker σ=kM:σk=eσkand Imσ=σk:kM.

Theorem 2.3.Assuming that σ:MNbe a partial group homomorphism and kM. Then, the following are given.

  1. σek=eσk.

  2. σk1=σk1.

  3. kerσisasubpartial group ofM.

  4. Imσisasubpartial group ofN.

  5. σMkisasubpartial group ofNσk.

  6. σ1Nekisasubpartial group ofM.

Proposition 2.4[7] Suppose M,Nbe partial groups and σ:MNbe a homomorphism of partial groups. Then the following conditions are satisfied:

  1. IfAis a partial subgroup of M, then σ(A) is a partial subgroup of N.

  2. If Bis a partial subgroup of N, then σ−1(B) is a partial subgroup of M.

Proposition 2.5.Let σ:MNbe a homomorphism of partial groups. Then, it is obtained that

σek=eσk,σk1=σk1forallkM.

Definition 2.14.A sub partial group T=Hpof M=Gpis named wide if the set of idempotent elements of Mis a subset of T=Hpand normal, written M. (if it is wide and kTk1Tfor all kϵM).It is also trivial that the set of idempotents elements of Mis a normal subgroup of Mand it is called the set of idempotents elements of Mthe trivial normal subpartial group of M.

Theorem 2.4.Assuming that Mbe a partial group and kM, then

  1. Mkis a maximal subgroup of Mwith identity ek.

  2. M=Mk:kM=Mek:ekisin the set of idempotents elements of M.

Theorem 2.5.If Mis a partial group, then the set of idempotents elements ofMis commutative and central.

Definition 2.15.For a partial group homomorphismσ:MNit is defined kerσ=kM:σk=eσkand Imσ=σk:kM. Also, σ:MNis named isomorphism if it is bijective.

Definition 2.16.σis named as idempotent separating if σek=σesimplies that ek=es,whereek,esare in the set of idempotent elements of Mfor a partial group homomorphism σ:MN.

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3. Partial normal subgroups

In group theory, normal subgroup plays an important role in the classification of groups and gives lots of algebraic results. Now, we will construct an analog definition for partial groups. Throughout Gp will denote the partial group. In this chapter, we should notice that if Gis a group, then Gis a partial group with fecte. Also, [8] in a group every element has a unique inverse, but in partial groups [7] for every element aG, we have Inva0because of that reason the identity element of the group differs from the identity element of the partial group. We can continue to work under these assumptions. From here on in, we will use the notation Gp for partial groups.

Definition 3.1 (Partial Conjugation Criteria). Let Gp be a partial group, the element gp.xp.gp−1 (or gp−1.xp.gp) is called partial conjugate of xp by gp for fixed gp,xpGp.

Theorem 3.1.Let Gp be a partial group and Np be a partial subgroup of Gp then following conditions are satisfied:

  1. Np is normal in Gp if and only if for all xpNp and gpGp we have gp−1.xp.gpNp

  2. Np is normal in Gp if and only if for every element of Np all partial conjugates of that element also lie in Np.

Proof:

  1. It comes from the definition of partial normal subgroup.

  2. (:) Let Np is partial normal subgroup in Gp. Then we need to show for every xpNp and fixed gpNp the partial conjugates gp−1.xp.gp lies in Np. Since gpNp then gpGp. Also, since Np is a partial normal subgroup of Gp gp−1.xp.gpNp, this gives the proof.

(:) Conversely, let for every element of Np all partial conjugates of that element lie in Np: Then it comes directly from partial subgroup definition Conclusion(s). It is preferable to include a Conclusion(s) section which will summarize the content of the book chapter.

Remark 3.1.Any partial subgroup HpGp has right and left congruence (equivalence) class that cannot be the same. But if left and right congruence classes are the same (i.e., for any xGp, Hp.x= x.Hp) then Hp is called as normal partial subgroup.

Theorem 3.2.Let Gp be a partial group and Np be a partial subgroup of a partial group Gp and so the following conditions are coincided:

Proposition 3.1.

  1. Np is a partial normal subgroup of a partial group Gp.

  2. gp.Np=Np.gp for all gpGp,

  3. gp.Np.gp−1Np for all gpGp.

Proof:

(i) (ii) If Np is a partial normal subgroup of Gp then it is easy that for all xpNp and gpGp we have gp.xp.gp−1Np and from just before the theorem gp.Np.gp−1Np.

(iii)(ii) Let gp be an element of Gp: We need to see gp.Np=Np.gp. Assume that xpgp.Np. Then xp = gp.n1 is satisfied for n1Np. Sincexp.gp−1 = gp.np.gp−1 and gp.Np.gp−1Np we have xpgp−1Npand so that there exists an n2Np such that xp.gp−1 = n2. If we product from right with gp then we have the (xp.gp−1).gp = n2.gp equality.

Using the associativity property the equality becomes xp.(gp−1.gp) = n2.gp and using identity element property and converse element property we get xp = n2.gpNp.gp. It implies that gp.NpNp.gp. In a similar way gp−1.Np.gpNp and we have Np.gpgp.Np. Therefore we obtain gpNp = Np.gp.

(ii)(i) Supposing that gpGp we have to prove gp.np.gp−1 is contained in Np for all npNp. Since gp.Np = Np.gp, we can say that gp.npNp.gp for all npNp. Then, by associativity gp.np.gp−1 is contained in Np.gp.gp−1 and then for all npNp, gp.np.gp−1.

Lemma 3.1.The center of a partial group Gp is a partial normal subgroup of Gp.

Proof:The center of a partial group is defined as below:

Ζ(Gp)={xpG│If for every gpGp xp.gp and gp.xp are defined, xp.gp=gp.xp}.

Let gpGp and xpΖ(Gp) then we need to show gp.xp.gp−1 is contained in Ζ(Gp). Since xpΖ(Gp) for every gpGp if gp.xp and xp.gp is defined then xp.gp = gp.xp. Using this argument gp.xp.gp−1=xp.gp.gp−1=xpΖ(Gp) and then we have Z(Gp) is a partial normal subgroup of Gp.

Proposition 3.2.Let φ: GpHp be a partial group homomorphism. Then the kernel of φis partial normal subgroup of Gp.

Proof:Let K = Ker(φ). We know that Ker(φ) is a partial subgroup of Gp. Suppose that yKand gpGp. Then using the fact φis a partial group homomorphism

φgp.yp.gp1=φgp.φyp.φgp1=φgp.eHp.φgp1=φgp.φgp1=eHpwe haveφgp.yp.gp1=eHp

and we get gp.yp.gp−1K. So Ker(φ)is a partial normal subgroup of Gp.

Partial normal subgroups

Theorem 3.3.Suppose Gp is a partial group and Hp is a partial subgroup of Gp.Hp is a partial normal subgroup of Gp if and only if (aHp) (bHp) = abHp equality holds for all a,bHp.

Proof:

(:) Suppose Hp is a partial normal subgroup of Gp. We need to see the equality (aHp) (bHp) = abHp holds.

(:) Let x(aHp) (bHp) then h,h2Hsuch that x= (ah1) (bh2). By using associativity property, we get x=(h1bh2). From the identity element, x = abb−1h1bh2 is obtained. Considering associativity property we can write x=ab(b−1h1b)h2. Since Hp is a partial normal subgroup of Gp, b−1h1bHp Then x abHp and this implies that (aHp) (bHp) abHp.

(:)Conversely, let yabHp then there exists an hHp such that y=abh. So we can write y = abhas follows; y=aebh(aHp) (bHp). It implies that abHp(aHp) (bHp). Therefore (aHp) (bHp)=abHp

(:) Let us consider (aHp) (bHp) =abHp for all a,bGp. If hHp and gGp then we must see whether or not ghg−1H.Using associativity property and considering hypothesis; ghg−1 = (gh)(g−1e) (gHp)(g−1Hp) = gg−1Hp = Hp. This implies that ghg−1Hp. So that Hp is a partial normal subgroup of Gp.

Theorem 3.4.Suppose Gp be a partial group and H,Kare partial subgroups of Gp. If Kis a partial normal subgroup of Gp, then the following cases are satisfied:

  1. HKis a partial normal subgroup of H.

  2. If Kand Hare partial normal subgroups of Gp and HK= ethen hk=kh(or, HK=KH) for every hHand every kK.

Proof:

  1. Since Hand Kare partial subgroups, HKis also a partial subgroup of G. By HKKwe can conclude that HKis also a partial subgroup of H. Let consider aHKand hH: Sincea,hH, ha and hah−1 can be defined. It gives that hah−1HAlso since Kis a partial normal subgroup of Gp we have hah−1K. It shows that HKis a partial normal subgroup ofH.

  2. Let Hbe a partial normal subgroup of Gp, HK=eand hHand kK. Since His a partial normal subgroup, we know that k.h.k−1H. If h.k.h−1.k−1Hthen we get (h.k.h−1).k−1Kby Kis a partial normal subgroup. So we have, h.k.h−1.k−1HK=e. Then h.k.h−1.k−1=eand so, hk=kh; for all hH,kK. Thus, we prove that HK=KH.

Proposition 3.3.Let Gp be a partial group and Hp be a partial subgroup of index 2 in Gp: Then Hp is a partial normal subgroup in Gp.

Proof:Let Hp be a partial subgroup of index 2 in Gp and gp be an element of Gp. If gpHp, then gpHp = Hpgp is satisfied. If gp is not in Hp, two left cosets must be as Hp and gpHp. Since left cosets are disjoint we know gp.Hp = GpHp. Also, the right cosets are disjoint so we can write Hp.gp = GpHp. Thus gpHp = Hpgp for all gpGp. So Hp is normal.

Example 3.1.Let Gp be an abelian partial group. Then any subgroup of Gp is a partial normal subgroup of Gp.

Proof:

If Gp is an abelian partial group and xp.ypGp thenxp.yp = yp.xp for every xp.ypGp. If gp.xp.gp−1Np thenNp is a partial subgroup of Gp. Using the hypothesis we get

gp.xp.gp1=gp.gp1.xp=eGp=xpNp

Therefore, the partial subgroupNpis the partial normal subgroup of Gp.

Example 3.2.Let Hp and Kp be any partial normal subgroup of Gp. Then Hp×Kp is also a partial normal subgroup of Gp×Gp.

Proof:

Hp×Kp = {np=(hp,kp) hpHp and kpKp}. We have to show that for all gp in Gp and np in Hp×Kp gp.np.gp−1 is in Hp×Kp.

gp.np=gp.hpkp=gp.hpgp.kp

and

gp.np.gp1=gp.hpgp.kp.gp1=gp.hp.gp1gp.kp.gp1

and since Hp and Kp are partial normal subgroups of Gp, then gp.hp.gp−1Hp, and gp.kp.gp−1Kp and so that (gp.hp.gp−1, gp.kp.gp−1)Hp×Kp. Then the Cartesian product of two partial normal subgroups is also a partial normal subgroup.

Theorem 3.5.Let be a partial group and Np be a partial normal subgroup of Gp. The congruence modulo Np is a congruence relation for the partial group operation “.”.

Proof.Let xRNpydenote that x and y are in the same coset, that is; xRNpyx.Np =y.Np

Let xRNpxand yRNpy. To demonstrate that RNp is a congruence relation for ., we need to show, reflexivity, symmetry, and transitivity. These axioms are obvious from the definition of relation.

Theorem 3.6.If Np is a partial normal subgroup of a partial group Gp and Gp=Np is the set of all cosets of Np in Gp, then GpNp is a partial group under the operation given by (aNp)(bNp)=abNp.

Proof.Let aNp, bNp,cNpGpNp. We must see partial group axioms are satisfied:

(G1) If (aNp)(bNp)=abNp. (bNp)(cNp)=bcNp and a.(bcNp) are defined then

a.bcNp=aNpbNpcNp=(aNpbNpcNp=abcNp

(G2) For any aNp in GpNp, eNp is a candidate for identity element, i.e.,

aNpeNp=aeNp=aNp

and

eNpaNp=eaNp=aNp

(G3) Since is a partial groupaGp has an inverse áGp. For every aNp in GpNpáNp is a candidate for the inverse of aNp.

aNpáNp=aáNp=eNp=Np

This completes the proof.

Definition 3.2.Let Gp be a partial group and Np is a partial normal subgroup of Gp, then the partial group GpNp is called the quotient of the partial group or factor group of Gp by Np.

Proposition 3.4.Letf: GpHp be a homomorphism of partial groups, then the kernel of fis a partial normal subgroup of Gp. Conversely, if Np is a partial normal subgroup of Gp, then the map ∏: GpGpNp given by (ap)=aNp is an epimorphism with kernel Np.

Proof:Kerf={xpGpf(xp)=eHp},we need to show that if xpKerfand apGp whether or not apxpap−1Kerfif and only if f(apxpap−1)=f(ap)f(xp) f(ap−1)=eHp

So, apxpap−1Kerf. Therefore, Kerfis a partial normal subgroup of Gp. It is trivial that ∏: GpGpNp is surjective. Ker={xp│∏(xp)=epNp}=Np.

Theorem 3.7.Let f:GpHp is partial group homomorphism and Np is a partial normal subgroup of Gp contained in the kernel of f, then there is exactly unique homomorphism f¯: GpNpHp such that f¯(aNp)=fafor all aGp. Besides Imf=Imf¯and kerf¯=(kerf)/Np.f¯an isomorphism if and only if fis an epimorhism and Np=kerf.

Proof:f:GpfHp, GpGpNp, GpNpf¯Hp diagram is commutative. If baNp, then b=anp,npNand also f(bp)=f(anp)=f(a)f(np)=fae=fa, since Npkerf. Therefore, fhas the same effect on every element of aNp and the map f¯: GpNpHp given by f¯(aNp)=fa. It is easily seen that f¯is a well-defined function. Now we need to prove whether or not f¯is a homomorphism of partial groups.

faNp.bNp=f¯abNp=fab
=fa.fb=f¯aNp.f¯bNp

So, fis a partial group homomorphism. Imf¯=Imfand aNpkerf¯fa=eakerf, whence

kerf¯=aNpakerf=kerfNp

f¯is unique since it is completely determined by f. Also, f¯is a partial group if and only if fis an epimorphism of partial groups f¯is a monomorphism if and only if for kerf¯=kerfNpkerfequal to Np.

Example 3.3.In Example 2.2, it is stated that G={0,±1,±2,…,±n} is a partial group with known addition operation on Z. We can easily say that the subset N= {0,±1,±,…,±n1}of Gis a partial subgroup of G. Let us show whether or not Nis a partial normal subgroup. If gp−1 + np + gpNfor all gpGp then Nis a partial normal subgroup. gp−1=gp in this group so that gp + np + gpNp i.e. npNand then Nis a partial normal subgroup of G.

Theorem 3.8 (First Isomorphism Theorem for Partial Groups).Let Gp, Hp be partial groups and f:GpHp be partial group epimorphism then GpKerfis isomorphic to Hp.

Proof:We know that kerfis a partial normal subgroup of Gp. Then GpKerfis defined. Let show Kerf=Kand g:GpKHp mapping defined as gaKbK=gabK=fabfor every aK,bKGpK.

Since ga.bK=fab=fafb=gaKgbKthen g is a homomorphism. Since fis onto there exists aGp such that fa=hthen aKGpKand gaK=fa=hand g is onto. For one-to-one conditions let aK,bKGpKand

gaK=gbKifffa=fb
fb1fa=fb1fbifffb1a=eHp
b1aKorkerfiffaK=bK

Then, g is an isomorphism and GpKerfHp.

Lemma 3.2.Let Gp be a partial group and Hp be a partial subgroup of Gp and Np be a partial normal subgroup of Gp. If Hp is a partial normal subgroup of Gp; then HpNp is a partial normal subgroup of Gp.

Proof:Trivial.

Theorem 3.9 (Second Isomorphism Theorem for Partial Groups).Let Gp be a partial group. Hp be a partial subgroup of Gp and Np be a partial normal subgroup of Gp, then the following isomorphism holds:

HpNpNpHpHpNp

Proof:It can be easily seen using First Isomorphism Theorem. So, the proof is left to the reader.

Theorem 3.10 (Third Isomorphism Theorem for Partial Groups).Let Gp be a partial group and Hp,Np partial normal subgroups of Gp with HpNp Then Hp is also a partial normal subgroup of Np.NHp is a partial normal subgroup of GpHp and also GpNp is isomorphic to (GpHp)(NpHp).

Proof:Let definef:GpHpGpNp,f(aHp)=aNp First let show fis well-defined: If aHp = bHp then we need to prove whether or not f(aHp)=f(bHp). Since aHp = bHp then ab−1Hp and HpNp,ab−1Np. Since ab−1Np then aNp = bNp and so that f(aHp)=f(bHp), i.e., fis well defined. fis homomorphism. And using the First Isomorphism Theorem we can conclude the result.

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4. Conclusion

There are some papers such as solvable partial groups, topological structures of partial group, Transitivity Theorem-Thompson Theorem of partial groups, k-partial groups, primitive pairs of partial groups so on related with the partial group in the literature. It is known that every group is partial but the converse is not true. That is why some structures are different from each other for the usual group and partial group.

In this chapter, some structures of partial groups (Clifford Semigroup) are sought to demonstrate algebraically. At the beginning of the chapter (preliminaries section), several fundamental results of partial groups with some numerical examples are given from the literature. For example, if A and B be two usual groups such that the intersection of them is equal to {1 = e}, then the union of subgroups of A and subgroups of B is a partial group. Partial normal groups and partial quotient groups have introduced an analog of the group theory. By using them, a number of isomorphism theorems are proved for partial groups with several other ideas. All results are obtained using closely group theory as algebraic approximations. Readers also may consider/investigate other structures/properties of the partial groups different from the group as algebraically.

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Conflict of interest

The author declare no conflict of interest.

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Written By

Özen Özer

Reviewed: December 20th, 2021 Published: February 22nd, 2022