Abstract
A picture in a group presentation is a geometric configuration with an arrangement of discs and arcs within a boundary disc. The drawing of this picture does not have to follow a particular rule, only using the generator as discs and the relation as arcs. It will form a picture label pattern if drawn with a particular rule. This paper discusses the label pattern of a picture in the presentation of direct product groups. Direct product presentation is used with two cyclic groups, and where and . The method for forming a picture label pattern is to arrange the first generator in the initial arrangement, compile a second generator, and add a number of commutators. Furthermore, the pattern is used to calculate the length of the label on the picture. It is obtained that the picture’s label is and the length of the label is , where is the number of commutator discs.
1. Introduction
Let be a group and is the presentation of , where is a set of generators and is a set of relations. A picture over is an object with a particular condition. Let be a picture over , and then there are various picture forms for , but with different labels.
The theory of identity sequences over group presentations is given by [1]. The article discusses algebraic theory and an elementary and complete exposition of the theory of pictures. Moreover, it gives some examples of so-called combinatorially aspherical presentations. The author in [2] discusses combinatorial geometric techniques that determine explicit generators for the second homotopy module of the 2-complex in terms of its cell structure. The discussion focuses on the theory of pictures from a homotopy-theoretic perspective and then generalities related to the generation of second homotopy modules with some proof of its properties. The article also gives various calculations and applications for studying second homotopy modules. Furthermore, this study was expanded by [3], i.e., the extensions of a group presentation by group presentation by adding reduced word on the generator of .
The generator’s second homotopy module is calculated by [4]. The first discussion is about the relationship between the second homotopy module for two presentations. And it is then defining an isomorphic group using the Tietze transformation. A study of the picture related to the group presentation was obtained by applying the Kronecker product to the representation of the quaternion group given by [5]. Meanwhile, [6] discusses the picture of the crossed product of groups and finds the generator of its second homotopy modules.
In this paper, the author tries to study differently, i.e., to form a pattern of the picture labels of the presentation of the direct product group. It is used as the direct product of two cyclic groups, and where and .
The paper aims to create the pattern of the picture through the presentation of direct product groups of two cyclic groups. In this case, the pattern of the picture’s label in the presentation of a cyclic group is associated with the number of arcs connected to the boundary disc.
The organization of this paper is as follows: the Preliminaries section introduces a lot of basic concepts and notations of a word, the presentation of the group, and the theory of picture, which will be used in the Result and Discussion section. The Result and Discussion section discusses the shape of the picture pattern in the presentation of the direct product by giving a formula for the picture in the presentation of a cyclic group that is associated with the number of arcs connected to the boundary disc.
Since the location of the arcs on the disc in a picture of the presentation of a cyclic group can be anywhere as much as the order of the cyclic group, it takes work to form a pattern in the picture. Thus, this paper has proven the properties of the picture in the presentation of the direct product (Theorem 12 and Corollary 13).
2. Preliminaries
We start the notion with the word, presentation groups, and picture over presentation groups.
A group presentation is pair , where is a set of generators and is nonempty, cyclically reduced word on (the relators).
Definition 1 (see [7]). Let be a set of distinct elements and . Define . A word is a finite string with the form , , , , and .
There are four operations on words on , that is (i)If contains a subword , , and , then delete it(ii)Insert a word , , and at any position on (iii)If contains a subword , , and , then delete it(iv)Insert , , and at any position in
The set of words will be denoted by . Two words, and , are equivalent (relative to ) if one of the words can be obtained from the other by finite operations on words. If and are equivalent, then they are symbolized by . Relation is an equivalence relation in .
Let be the equivalence class containing , then a set of equivalence classes with binary operation for every form a group. The group defined by denoted by with identity in is and . If is understood to be a group presentation for , then we refer to simply .
Two words, and , are equivalent (relative to ) if one of the words can be obtained from the other by finite operations on words. If and are equivalent, then they are symbolized by . The relation is an equivalence relation on , and is the equivalence class containing . Set with binary operation form a group, i.e., the free group containing as a basis, denoted by . We usually dispense with equivalence class notation and write for . If , then we write and say and are freely equivalent.
We have the following properties:
Theorem 2 (see [8]). If is any set, there is a free group having as a free basis.
Lemma 3 (see [9]). If is free on , then generates .
Theorem 4 (see [10] Characterization of freeness). Let be a group and be a subset of . Then is free with basis if and only if the following both hold: (1) generates , and(2)If is a word on and , then is not freely reduced, that is, must contain an inverse pair
Definition 5 (see [2]). Let be a group presentation. A picture over is a geometric configuration consisting of the following: (i)A boundary disc with a basepoint(ii)Discs with labels read an element of (clockwise or anticlockwise)(iii)Disjoints arcs with label elements of
The picture can be illustrated in Figure 1.
The picture over is spherical if it has at least one disc, and no arc of meets the boundary disc.
A picture over becomes a based picture over when it is equipped with a basepoint as follows: (i)Each disc has one base point, a selected point in the interior of a basic corner(ii)Picture has a global base point, a selected point in the boundary disc that does not lie on any arc of
Two pictures will be equivalent if one can be transformed to the other by a finite number of delete/insert floating circle, delete/insert canceling pairs, bridge move, and replace(), where be a set of based spherical pictures over .
Let be a picture over labeled by , then the length of , symbolized by , defined as the arcs connected to the boundary disc that is read clockwise.
Definition 6 (see [7]). Let be a cyclic group with the generator , . Then, the presentation of a cyclic group order is defined as .
Proposition 7 (see [7]). A presentation of a cyclic group is isomorphic to , ().
Definition 8 (see [7]). Let and be groups. The direct product of and is the set of all ordered pairs with the operation .
Theorem 9 (see [9]). Let and be groups defined by presentations and , respectively. Then, the presentation defines the presentation of the direct product group .
Based on Definition 8, we have a presentation of : let and be the presentation for and , respectively, then and or
3. Results and Discussion
To form the picture pattern of , the following rules are used: (1)Establish the desired picture pattern by setting the first disc’s location as disc (and continuing another disc on the right side as needed)(2)Each number of disc and disc is one(3)Picture pattern based on the number of discs
The picture is introduced in the presentation of a cyclic group with two forms, i.e., a picture with one lane and two lanes.
The following theorem discusses the number of arcs connected to the boundary disc for the picture of the presentation of a cyclic group.
Theorem 10. Let be a picture over and with the number of disjoint discs and , and every disjoint disc is connected with one arc (see Figures 2 and 3). Then, the number of arcs connected to the boundary disc is as follows: (a), , , and is a picture with a one lane(b), is even, and is a picture with two lanes
Remark 11. (i)For , we have two arcs connected to the boundary disc for a picture with one lane, and there is no picture with two lanes(ii)For , we have arcs connected to the boundary disc, where for a picture with one lane and the number of arcs connected to the boundary disc is , where is even for a picture with two lanes, and there is no picture with two lanes, where is odd
Proof of Theorem 10. (a)The proof is divided into two cases, for is odd and even (i) is oddConsider that , , and are disjoint discs in picture and is a symbol for the number of arcs from . It is found that and are , and . Thus, , so it is true for . Suppose it is true for , is odd, we have . So, it is true for . (ii) is evenConsider that , , , and are disjoint discs in picture . It is found that and are and . Thus, , so it is true for . Suppose it is true for , is even, we have . (b)Since is a picture with two lanes, the number of discs in is even. Consider that , , , and are disjoint discs in picture . It is found that for Thus, . So, it is true for . Suppose it is true for , is even, we have So, it is true for Since is a picture with two lanes, the number of discs in is even.
Theorem 12. Let be a picture over , where . Then, picture has the label , where is the number disc of the commutator and .
Proof. The proof is divided into two cases: for and for .
Case . We have to consider the following picture.
Let be a picture over . Picture has three types of discs, i.e., discs with arcs (), discs with arcs , and discs with arcs ().
For , by taking the basepoint as in Figure 4.
The number of arcs connected to the boundary disc is obtained.
(i)The number arcs on are (ii)There is one arc on each disc for ( is the number of commutators)(iii)The number of arc on for is
For more details, it can be seen in Table 1.
Note that the label for picture can be written in the form for . Furthermore, with the operation on the picture and a transformation on the picture, any picture with any determination of base points is equivalent to the picture labeled (see [4]).
Case
For , consider the picture in Figure 5.
Assume that the base point as in Figure 5, the number of arcs connected to the boundary disc is (i)The number arc on is (ii)In the opposite direction, there is one arc on , respectively, for , and two arcs , respectively, for (iii)No arc connected to the boundary disc on for
For more details, it can be seen in Table 2.
Thus, the label of picture is for . Furthermore, with the operation on the picture, any picture over is equivalent to the picture labeled by .
Corollary 13. Let be a picture over , where . Then
Proof. Let be a picture over . Based on Theorem 12, any picture over has the label . So for and for .
4. Conclusions
This article provides a picture pattern of the presentation of the direct product of two cyclic groups by setting a particular disc position at the beginning; in this case, the disc . The new picture pattern with this condition is given. Based on this pattern is one way to determine the label of the picture of the presentation.
Data Availability
The data used to support the findings of this study are included in the article.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This research was funded by the Andalas University (Grand No. T/34/UN.16.17/PT.01.03/IS-RPB/2021)