Artin Characters for the Special linear group Design an Algorithm to Calculate the Inverse Permutations Of the Symmetric Group Sn In Computer Hani M. Abood Al-Dainy Dept. of Computer Sci./ College of Education for pure Science (Ibn Al-Haitham)/ University of Baghdad Received in: 24 March 2014 , Accepted in:14 April 2014 Abstract The search is an application for one of the problems of mathematics in the computer; as providing construction and design of a major program to calculate the inverse permutations of the symmetric group Sn , where 1 ≤ n ≤ 13; using some of the methods used in the Number Theory by computer . Also the research includes design flow chart for the main program and design flow chart for the program inverse permutations and we give some illustrative examples for different symmetric groups and their inverse permutations. Keywords: Symmetric group, permutations, inverse permutations, identity permutation, length of cycle. 385 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Introduction Number Theory is concerned with properties of the integers: . . . ,−4,−3,−2,−1, 0, 1, 2, 3, 4, . . . . The great mathematician Carl Friedrich Gauss called this subject arithmetic and he said: (1) “Mathematics is the queen of sciences and arithmetic the queen of Mathematics” Symmetry property can be described as many things, such as objects of engineering and mathematical equations, etc., and it is characterized by symmetry recipe Human Man convicted him, two hands, two legs, two eyes and ears, this means that half of it is similar to the right half of the left form. In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G (which are thought of as injective functions from the set M to itself); the relationship is often written as (G,M). Note that the group of all permutations of a set is the symmetric group; the term permutation group is usually restricted to mean a subgroup of the symmetric group. The symmetric group of n elements is denoted by Sn; if M is any finite or infinite set, then the group of all permutations of M is often written as Sym(M).(2) The application of a permutation group to the elements being permuted is called its group action; it has applications in both the study of symmetries, combinatory and many other branches of mathematics, physics and chemistry. The degree of a group of permutations of a finite set is the number of elements in the set. At first blush one might think that of all areas of mathematics certainly Arithmetic should be the simplest, but it is a surprisingly deep subject. Mathematics is of abstract science, but do not touch reality only when we see its applications. So one of the Mathematics problems we applied in the computer. This is guaranteed by our research. The inverse of permutation for Sn has been given by Morris (3) for 1 ≤ n ≤ 13. It seems to be that there are no-ready-programs available to calculate those tables. We give algorithms of programs to calculate the inverse for Sn 1≤ n ≤ 13. In this paper we adopt the properties of inverse and permutations for Sn. Definition: (1) (4) Let X is a non-empty set, a permutation of X is a function α: X → X that is a one-to- one correspondence and the symmetric group on X, denoted by SX, is the group whose elements are the permutation of X and whose binary operation is composition of function, of particular interest is the special case when X is finite. If X = {1,2,3,…,n}, we write Sn instead of SX and we call Sn the symmetric groups of degree n, or the symmetric groups on n letters. Note that Sn = n! Definition: (2) (5) A permutation 1 2 3 k 2 3 4 1 x x x ... x α x x x ... x   =     of a set X is called a cycle of length k or a k-cycle written α = (x1x2x3…xk) and α(x) = x for all x ∈ X but x ∉ {x1,x2,x3,…,xk} k is called the length of the cycle. Definition: (3) (3) The product of two cycles need not again be a cycle. Thus α = (x1x2 … xk) then α– 1 = (x1xk … x2) such that α. α– 1 = α– 1 .α = i (identity permutation). From the previous definitions we can deduce the following properties: (1) A cycle of length 2 has inverse that same cycle. (2) The inverse of identity permutation is itself. 386 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 http://en.wikipedia.org/wiki/Group_(mathematics) http://en.wikipedia.org/wiki/Permutation http://en.wikipedia.org/wiki/Set_(mathematics) http://en.wikipedia.org/wiki/Bijective_function http://en.wikipedia.org/wiki/Symmetric_group http://en.wikipedia.org/wiki/Subgroup http://en.wikipedia.org/wiki/Group_action http://en.wikipedia.org/wiki/Group_action http://en.wikipedia.org/wiki/Symmetry http://en.wikipedia.org/wiki/Combinatorics http://en.wikipedia.org/wiki/Mathematics http://en.wikipedia.org/wiki/Finite_set http://en.wikipedia.org/wiki/Cardinality (3) A cycle has two kinds joint and disjoint. The Algorithms: This part contains a collection of the computer algorithms for many standard methods of Number Theory installed in our main program. Algorithm (1): The Length of Cycle This algorithm is designed for determining the length of Cycle of permutation for Sn. Input: Sn (The symmetric group of degree n) Step 1: To evaluate k If α = (a1 a2 … ak) ∋ α(a) = a for a ∈ ai But a ∉ {a1 a2 … ak} Then k is length of cycle Output: The length of cycle (k) End. Algorithm (2): The Inverse of Joint Cycle This algorithm is designed for determining the inverse of joint cycle. Input: Sn (The symmetric group of degree n) Step 1: k1 is cycle of length k1,(By using algorithm 1) Step2: For I = 1 to k1 ,A(I) is inverse of cycle of length I B (1) = A (1) B (2) = A (k1) End Step 3: For J = 2 to k, L = 0 B (J) = A (k – L) L = L + 1 Print B (J) End J-Loop Output: The inverse of joint cycle. Algorithm (3): The Inverse of Disjoint Cycle This algorithm is designed for determining the inverse of disjoint cycle. Input: Sn (The symmetric group of degree n) Step 1: let 2 and (n – 2)! are two disjoint cycles Step 2: For I = 1 to 2 A1 (I) Print A1 (I) End I-Loop Step 3: For I2 = 1 to (n – 2)! A2 (I2) Print A2 (I2) End I2-Loop Step 4: B1 (1) = A1 (1) L1 = 0 For J1 = 1 to 2 B (J1) = A (2 – L) L1 = L1 + 1 Print B (J1) End J1-Loop Step 5: B2 (1) = A2 (1), L2 = 0 For J2 = 1 to (n – 2)! B (J2) = A2 ((n – 2)! – L2) L2 = L2 + 1 387 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Print B2 (J2) End J2-Loop Step 6: B1 (J1) B2 (J2) The inverse of cycle of length 2 is B1 (J1) and the inverse of cycle of length (n – 2)! Is B2 (J2), that the inverse of disjoint cycle is B1 (J1) B2 (J2). Step 7: If k1 = 3 and k2 = (n – 3)! Then in general k & ( n – k)! Output: The inverse of k & (n – k)! Algorithm (4): Special Case the Degree of the Sn . Input: Sn (The symmetric group of degree n) Step 1: For I = 1 to n Print A (I) If A (I) = I then A (I) is inverse of cycle of length I Else End I-Loop Step 2: For J = 1 to n Print A (J) If (J.GE.2) then B(J) = INDEX(I) End J-Loop Output: The inverse of identity and cycle of length 2 End Algorithm (5): The main Algorithm 1 ≤ n ≤ 13. Input: Sn (The symmetric group of degree n) Step 1: Call algorithm 1 Step 2: Call algorithm 2 Step 3: Call algorithm 4 Step 4: Call algorithm 3 Output: (IN (k)) (To evaluate the inverse of permutation for Sn) End 388 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Flow Chart of the Main Program 389 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 The following will give some illustrative examples of the different symmetric groups and their inverse permutations. Example : (1) The elements of S4 and the inverse permutations of the S4 elements Elements of S4 Inverse permutations of the S4 elements 1 (1234) (1432) 2 (2341) (2143) 3 (1432) (1234) 4 (1324) (1423) 5 (1423) (1324) 6 (1342) (1243) 7 (123) (132) 8 (342) (432) 9 (143) (134) 10 (243) (234) 11 (132) (123) 12 (134) (143) 13 (142) (124) 14 (12) (21) 15 (23) (32) 16 (34) (43) 17 (41) (14) 18 (24) (42) 19 (13) (31) 20 (14) (41) 21 (43) (34) 22 (21) (12) 23 (31) (13) 24 i i 390 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Example: (2)The elements of S5 and the inverse permutations of the S5 elements Se. Elements of S5 Inverse Permutations of the S5 elements Se. Elements of S5 Inverse permutations of the S5 elements 1 i i 41 (143) (134) 2 (12) (21) 42 (153) (134) 3 (13) (31) 43 (154) (145) 4 (14) (41) 44 (243) (234) 5 (15) (51) 45 (253) (235) 6 (23) (32) 46 (254) (245) 7 (24) (42) 47 (345) (354) 8 (25) (52) 48 (35)(142) (53)(124) 9 (34) (43) 49 (12)(345) (21 )(354) 10 (35) (53) 50 (13)(245) (31 )(154) 11 (45) (54) 51 (14)(235) (41)(253) 12 (12)(34) (21)(43) 52 (15)(234) (51)(432) 13 (12) (35) (21) (53) 53 (23)(145) (32)(154) 14 (12) (45) (21) (54) 54 (24)(135) (42)(153) 15 (13)(45) (31)(54) 55 (12)(354) (21)(345) 16 (23)(45) (32)(54) 56 (13)(254) (31 )(245) 17 (13)(24) (31)(42) 57 (14)(253) (41)(235) 18 (13)(25) (31)(52) 58 (15)(243) (51)(234) 19 (14)(25) (41)(52) 59 (23)(154) (32)(145) 20 (14)(35) (41 )(53) 60 (24)(153) (42)(135) 21 (24)(35) (42)(53) 61 (25)(134) (52)(143) 22 (14)(23) (41)(32) 62 (34)(125) (43)(152) 23 (15)(23) (51)(32) 63 (35)(125) (53)(142) 24 (15)(24) (51)(42) 64 (35)(124) (53)(142) 25 (15)(34) (51)(34) 65 (45)(132) (54)(123) 26 (25)(34) (52)(43) 66 (1254) (1452) 27 (123) (132) 67 (1354) (1453) 28 (124) (142) 68 (2354) (2453) 29 (125) (152) 69 (1342) (1243) 30 (134) (143) 70 (1352) (1235) 31 (135) (153) 71 (1452) (1254) 32 (145) (154) 72 (1453) (1354) 33 (243) (234) 73 (2453) (2354) 34 (235) (253) 74 (1432) (1234) 35 (245) (254) 75 (1532) (1235) 36 (132) (123) 76 (1542) (1245) 37 (142) (124) 77 (2543) (2345) 38 (152) (125) 78 (1543) (1345) 39 (143) (134) 79 (12345) (15432) 40 (152) (152) 80 (12354) (14532) 391 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Se. Elements of S5 Inverse permutations of the S5 elements Se. Elements of S5 Inverse permutations of the S5 elements 81 (12435) (15342) 101 (34)(152) (43)(125) 82 (12453) (13542) 102 (12543) (13542) 83 (1243) (1342) 103 (12543) (15342) 84 (]253) (1325) 104 (13254) (13542) 85 (1253) (1352) 105 (13425) (14352) 86 (1254) (1452) 106 (13452) (13452) 87 (]354) (1453) 107 (13524) (13452) 88 (2354) (2453) 108 (13542) (14523) 89 (1342) (1243) 109 (14235) (12543) 90 (1352) (1235) 110 (14253) (14253) 91 (1452) (1254) 111 (14325) (12453) 92 (1453) (1354) 112 (14352) (15324) 93 (2453) (2354) 113 (14523) (13524) 94 (1432) (1234) 114 (14532) (15234) 95 (1532) (1235) 115 (15234) (12534) 96 (1542) (1245) 116 (15243) (13254) 97 (2543) (2345) 117 (15324) (14325) 98 (1543) (1345) 118 (15432) (13425) 99 (1534) (1435) 119 (15342) (14235) 100 (25)(143) (52)(134) 120 (15423) (13245) Conclusions 1. Algorithms contained in the search provide the time and effort in the calculation of the inverse permutations of the symmetric group, as well as they give us the accuracy in the calculation. 2. The interval proposed in the search for the symmetric group Sn is 1 ≤ n ≤ 13; Therefore, the study the symmetric group Sn when n>13 analog is not without complexity. References 1-Clark,W. Edwin (2003), Elementary Number Theory, South Florida University. 2- Marcus du Sautoy (2009), Finding Moonshine: a Mathematician's Journey through symmetry ” Fourth Estate. 3- Morris,A.O. (1992), The Spin Nepresentation of the Sy Group, Proc. London, Math. Soc. (3), 12, 55-76. 4- Fraleigh ,J.B. (1982), A First Course in Abstract Algebra, Wesley Publishing Company. 5-Rotman, Joseph, J. (1988), An Introduction to the Theory of Groups, WM, C. Brown Publishers. 392 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 http://en.wikipedia.org/wiki/Marcus_du_Sautoy http://en.wikipedia.org/wiki/Fourth_Estate الحاسوب) Sn( تصمیم خوارزمیة لحساب معكوس التبادیل للزمرة التناظریة ھاني مسلم عبود الدایني جامعة بغداد /)ابن الھیثم (كلیة التربیة للعلوم الصرفة / قسم علوم الحاسبات 2012نیسان 14قبل البحث في : ، 2014اذار 24 استلم البحث في : الخالصة یمثل البحث تطبیقا" إلحدى مسائل علوم الریاضیات في الحاسوب ؛ إذ نقدم بناء و تصمیم برنامج ؛ باستعمال بعض الطرائق n ≤ 13 ≥ 1 ؛ إذ Snرئیس لحساب معكوس التبادیل للزمرة التناظریة مج الرئیس و تصمیم كذلك تضمن البحث تصمیم تخطیط للبرناالمعتمدة من نظریة األعداد في الحاسوب. مخطط لبرنامج معكوس التبادیل؛ وبعض األمثلة التوضیحیة لزمر تناظریة مختلفة ومعكوسات تبادیلھا. الكلمات المفتاحیة: الزمرة التناظریة ؛ التبادیل؛ معكوس التبادیل؛ التبدیل المحاید؛ طول الدورة. 393 | Computer Science @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Abstract The search is an application for one of the problems of mathematics in the computer; as providing construction and design of a major program to calculate the inverse permutations of the symmetric group Sn , where 1 ≤ n ≤ 13; using some of the me... Definition: (1) (4) Let X is a non-empty set, a permutation of X is a function (: X (( X that is a one-to-one correspondence and the symmetric group on X, denoted by SX, is the group whose elements are the permutation of X and whose binary operation is compositio... Note that (Sn( = n! Definition: (3) (3) استلم البحث في : 24 اذار 2014 ، قبل البحث في : 14 نيسان 2012