COMPUTING GENERATOR OF SECOND HOMOTOPY MODULE AND USING TIETZE TRANSFORMATION METHODS

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1 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 M 29 COMPUTING GENERATOR OF SECOND HOMOTOPY MODULE AND USING TIETZE TRANSFORMATION METHODS Yanita Department of Mathematics. Faculty of Mathematics and Natural Sciences. Andalas University, Kampus Unand Limau Manis Padang 25163, Indonesia yanita3010@gmail.com Abstract This article discuss about presentation group and. It is shown that these presentations are isomorphism and there is process to compute generator of second homotopy module from to. This computation using Tietze transformation and operation on picture. Keywords presentation group, second homotopy module, generator, Tietze transformation I. INTRODUCTION Group are very often described as quotient group of free group. If is free with base and is normal closure in of a set, we say that the pair is a presentation,. Set is defining generator of and set is defining relations. The element of will be called relator. A presentation is finitely generated if is finite, and is finitely related if is finite. A presentation is finite if both of and are finite, in this case is finitely presented. There are some alterations one can make to a presentation which result in presentations of a group isomorphic to the original. These are called Tietze transformations. Let so we have first fundamental group ( ) and second homotopy module (( of presentation group. Therefore this article discuss about second homotopy module. The element of second homotopy module is equivalence class of spherical picture. II. PRELIMINARIES We review some definitions and results that we will use to solve the main result of this article. Definition 2.1 (Definition of Tietze Transformation) ([3], [5]) Let and be two presentations of the group. (T1) If the word S is derivable from, then add S to the list of relators; (T2) If the word S is derivable from, remove S from the list relators; (T3) If R is word in the x, and y is some symbol not in the generating set, add y to the generating set and add word, to the relator set. (T4) If there is a relator of the form, with y not appearing in R, delete this relator and delete y from the generating set, replacing all order occurences of y in the relator words with. Theorem 2.2 ([5]) Suppose that the groups presented by the two presentations and are isomorphic. Then there is a sequence of Tietze transformations leading from one of these to the other. If these presentations are both finite the sequence can be taken to be a finite number of single step. Definition 2.3 ([6]) A picture over is a geometric configuration consisting of of the following M - 193

2 ISBN a. A disc with basepoint on. b. Disjoint discs in the interior of. Each has a basepoint on. c. A finite number of disjoint arcs where each arc lies in the closure of and is either simple closed curve having trivial intersection with, or is a simple non-closed curve which join two points of, neither point being a basepoint. Each arc has a normal orientation, indicated by a short arrow meeting with the arc transversely and is labelled by an element of. d. If we travel around once in clockwise direction starting from and read off the labels on arcs encountered (if we cross an arc, labelled say, in the direction of its normal orientation, then we read ), then we obtain a word which belongs to. We call this word the label of. Definition 2.4 ([6]) A picture over is a spherical picture if all arcs in do not touch the boundary disc. Two spherical pictures P 1 and P 2 are said to be equivalent if either (a) both are spherical and one can be transformed to the other by a finite number of operation deletion and insertion floating circle, deletion and insertion folding pair and bridge move; or (b) both are not spherical and one can be transformed to the other by a finite number of operation deletion and insertion floating circle, deletion and insertion semicircle, deletion and insertion folding pair and bridge move. Base on [1], A set of spherical picture over is called a set of generator if generate module. Base on [2], set generator P is generator iff each spherical picture over can be transformated to empty picture by using operation on picture. III. THE MAIN RESULT We consider presentation group and. It is shown that these presentation are isomorphic using Tiestze transformation. Note that, if and nonzero integer, there are integer and such that. Since and are prime relative, so.. Tietze transformation of to. Add generator in to set of generator with relation. Add in to set of relation, since derived from, and, that is. Add relator since derived from since since since Add relator since derived from since since since M - 194

3 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 Delete from set of relations, since and, so we have. Delete from set of relations, since and so we have. Delete from set of relations, since and so we have. Delete relator since derived from and, that is Delete generator from set of generators Delete generator from set of generators So we have. Base on [2] we have generators of are spherical picture containing discs,, and ; and spherical picture containing discs and disc. We say that generators are dan, respectively, i. e Figure 1 The generator of Whereas generator of is spherical picture containing disc, that is Figure 2 The generator of Furthermore, we use Theorem 1, Theorem 2, Corrolari 1 and Corrolari 2 on [7] to compute generator of second homotopy module of each presentation, i.e M - 195

4 ISBN Generators of equal to generator of, that is, dan. 2. Generators of are and, which is generator containing disc, and is picture containing discs,, and. 3. Generators of are dan, which is generator containing disc, that is and is picture containing discs, and. 4. Generators of are and, which is generator containing disc, M - 196

5 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 and is picture containing discs, and. 5. On there is a deletion disc and replace arc be and arc be. So, generators containing disc will be changed, as seen below 6. On there are deletion of disc. The generators containing disc will be changed, as seen below 7. On there is a deletion of disc The generators containing disc will be changed, as seen below M - 197

6 ISBN On there is a deletion. The generators containing disc will be changed, as seen below 9. On there is deletion generator. The generators containing disc will be changed, as seen below 10. On there is deletion generator. The generators containing disc will be changed, as seen below Finally, we have generator containing disc, that is Thus, generaotos of is M - 198

7 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 IV. EXAMPLE Consider group presentation and Remember that 2 and 3 are relatively prime, so there are integers and such that. Let and. Tietze transformation from to. We have four generators of, that is Meanwhile, generator of is M - 199

8 ISBN Next, will be shown the process of changing generators of to The generators of are,, and. The generators of are,,, and, where is 3. The generators of are,,,, and, where is 4. The generators of are,,,,, and, where is M - 200

9 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY Generators,,, and containing will be changed, and each one is named,,, and. M - 201

10 ISBN Generators of are,,,,, dan. M - 202

11 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY Generators containing are,,, and will be changed and each one is named,,, dan. M - 203

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13 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 Generators of are,,,, and 7. Generators,,, and will be changed and each one is named,,, dan. M - 205

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15 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 Generators of are,,,,,dan. 8. Generators, and will be changed and each one is named, dan. M - 207

16 ISBN Generators of are,,,,, dan. 9. Generators,,,, and will be changed and each one is named,,, dan. M - 208

17 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 M - 209

18 ISBN Generators of are,,,,, and. 10. Generators,,,, dan will be changed and each one is named,,,, and. M - 210

19 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 =. =. M - 211

20 ISBN If generator simplified, then we obtain picture By using bridge move operation, we have M - 212

21 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 2016 This picture same as picture This picture same as picture M - 213

22 ISBN Thus, we have generator of is REFERENCES [1] Y. G, Baik,., J. Harlander., S. J. Pride, The geometry of group extensions. J. Group Theory 1, No. 4, [2] W. A. Bogley, S. J. Pride, Calculating generators of. In Two-dimensional homotopy and combinatorial group theory (eds. C. Hog-Angeloni, W. Metzler & A. J. Sieradski), London Math. Soc. Lecture Note Ser. No. 197 (Cambridge University Press), pp [3] D. L.Johnson, Presentation of Group. Second Edition. London Mathematical Society, Student Text 15. Cambridge University Press. [4] W. Magnus, A. Karras, & D. Solitar Combinatorial Group Theory Presentations of Groups in Terms of Generator and Relations. New York Dover Publications, Inc. [5] C. F. Miller III, Combinatorial Group Theory. Lecturer notes on University of Melbourne. [6] S.J.Pride, Identity among relations of group presentations. In Group theory from a geometrical viewpoint (ed. E. Ghys, A. Haefliger and A. Verjovsky), (World Scientific), pp [7] Yanita & A.G. Ahmad, Computing Generators of Second Homotopy Module Using Tietze Transformation Methods. International Journal of Contemporary and Mathematical Sciences. Vol. 8 No M - 214