International Journal of Statistics and Applied Mathematics 2018; 3(2): 652-663 ISSN: 2456-1452 Maths 2018; 3(2): 652-663 © 2018 Stats & Maths The structure of the lattice of subgroups of the www.mathsjournal.com Received: 20-01-2018 symmetric group s5 Accepted: 21-02-2018 A Vethamanickam A Vethamanickam and C Krishna Kumar Associate Professor, Department of Mathematics Rani Anna Government College Abstract for Women, Tirunelveli, In this paper we determine all the subgroups of the symmetric group S5 explicitly by applying Lagrange’s Tamil Nadu, India theorem, Sylow’s theorem. We give the diagram of the lattice of subgroups of S5. We also study some lattice theoretic properties of the lattice of subgroups of Sn, when n≤5. C Krishna Kumar Assistant Professor, Keywords: Symmetric group, Lagrange’s theorem, Sylow’s theorem, p-Sylow subgroup, lattice Department of Mathematics, St. Jerome’s College of Arts and 1. Introduction Science, Ananthanadarkudy, Nagercoil, Tamil Nadu, India In mathematics, the notion of permutation relates to act of arranging all the members of a set into some sequence or order or the set is already ordered or rearranging its elements a process called permuting. For any non-empty set S, define A(S) be the set of all bijections (permutations) on S. A(S) together with the binary operation of composition of functions is a group. If the set S contain n elements, then the set A(S) is denoted by S . The group S has n! Elements and is called the n n symmetric group of degree n. Lattice of subgroups of S2, S3 and S4 have been studied already [2] [3] [5] [12] eg. Refer to , , and . In this paper we determine all the subgroups of S5, and then draw the diagram of the lattice subgroups of S5, also study some lattice theoretic properties. For the preliminaries and the basic definitions we refer to [2], [3], [4], [5] and [8]. 2. Preliminary We recall some definitions and results that will be used later. Definition 2.1 The symmetric group S is defined to be the group of all permutations on a set of five 5 elements, ie, the symmetric group of degree five. In particular, it is a symmetric group of prime degree and it is denoted by S5. Definition 2.2 A group generated by a single element is called cyclic and we know that cyclic groups are abelian. Definition 2.3 A subgroup of G is said to be a normal subgroup of G if for every g∈G and n∈N, gng-1∈N. Definition 2.4 A partial order on a non-empty set P is a binary relation denoted by ≤ on P that is reflexive, anti- symmetric and transitive. If ≤ is a partial order on P, the pair (푃, ≤) is called a partially ordered set or poset. A Poset (푃, ≤) is called totally ordered or a chain. If any two elements x, Correspondence C Krishna Kumar y ∈ P are comparable, that is either x ≤ y or y ≤ x. A non-empty subset S of P is a chain in P, Assistant Professor, if S is totally ordered by ≤. Department of Mathematics, St. Jerome’s College of Arts and Science, Ananthanadarkudy, Nagercoil, Tamil Nadu, India ~652~ International Journal of Statistics and Applied Mathematics Definition 2.5 Let (푃, ≤) be a poset and let S⊆P. An upper bound for S is an element x ∈ P for which s≤x for all s ∈ S. The least upper bound of S is called the supremum or join of S. A lower bound for S is an element x ∈ P for which x ≤ 푠 for all s∈ S. The greatest lower bound of S is called the infimum or meet of S. Poset (푃, ≤) is called the lattice if every pair x, y elements of P has a Supremum and an Infimum, and denoted by x∨y and x∧y respectively. Definition 2.6 The set of all subgroups of a given group G is a partially ordered set under ‘⊆’ inclusion relation. It is also a lattice denoted by L (G). Definition 2.7 In the poset (P,≤), a covers b or b is covered by a (in notation, a≻b or b≺a) if and only if b < a and, for no x ∈ P, b < x < a. Definition 2.8 Let G be a finite group and let |퐺| = pnm where n≥1, and p is a prime number such that (p, m) = 1. The subgroup of G of order pn is called a p-sylow subgroup of G. Definition 2.9 An element ‘a’ is an atom, if a≻0 and a dual atom, if a≺1. Definition 2.10 A lattice L is said to be modular lattice, if a ∨ (b ∧ c) = (a ∨ b) ∧ c for every a, b, c in L and a ≤ c. Definition 2.11 A Lattice L is said to be semi-modular if whenever a covers a ⋀ b, then a ⋁ b covers b, for all a, b ∈ L. Definition 2.12 An element of a lattice L is called join irreducible if x ⋁ y = a ⇒ x = a or y = a. Definition 2.13 A Lattice L is said to be consistent if whenever j is a join-irreducible element in L, then for every x in L, x ⋁ j is join irreducible in the upper interval [x, 1]. Theorem 3.1 If G is a finite group and H is a subgroup of G, then the order of H is a divisor of the order of G. Theorem 3.2 If G is a finite group and a ∈ G, then order of a is a divisor of order of G. Theorem 3.3 Let G be a finite group and let p be any prime number that divides the order of G. Then G contains an element of order p. Theorem 3.4 If P is a prime number and 푝훼 ∖o (G), 푝훼+1 ∤o (G), then G has a subgroup of order푝훼. Theorem 3.5 If G is a finite group, p is a prime number and 푝푛 ∖o (G), but 푝푛+1 ∤o (G), then any two subgroups of G of order pn are conjugate. Theorem 3.6 The number of p-sylow subgroups in G, for a given prime p, is of the form 1+kp. Theorem 3.7 If a group G has a unique sylow p-subgroup, then it is normal. Theorem 3.8 Let G be a group of order pq, where p and q are distinct primes and p<q. Then G has only one subgroup of order q and this subgroup of order q is normal in G. Theorem 3.9 If N is a normal subgroup of G and H is any subgroup of G, then NH is a subgroup of G. Theorem 3.10 If M and N are two normal subgroups of G, then NM is also normal subgroup of G. ~653~ International Journal of Statistics and Applied Mathematics Theorem 3.11 Sn has a normal subgroup of index 2 the alternating group An, consisting of all even permutation. Notation If G is a group, then we denote the lattice of all subgroups of G by the notation L (G). Result -1 A non-empty subset H of the group G is a subgroup of G iff a, b ∈ H⇒ab∈H and a ∈H ⇒ a ∈H. We produce below the structure of the lattice of subgroups of the symmetric groups S2, S3 and S4. Fig 1: Lattice Structure of L (S2) Fig 2: Lattice Structure of L (S3) Fig 3: Lattice Structure of L (S4) ~654~ International Journal of Statistics and Applied Mathematics 4. Main Result The symmetric group S5 is the group of all permutations of the set S = {1, 2, 3, 4, 5}, we know that the order of S5 is 120. We name the elements according to their orders. Let S5 = {i, 훼1, 훼2,… …, 훼10, 휎1, 휎2,… … , 훼20, 휏1, 휏2,……, 휏30, 훾1, 훾2……,훾15, 훽1, 훽2,……, 훽24, 훿1, 훿2,……, 훿20}, where ‘i’ is The identity permutation. The cycles of length 2 are, 훼1=(4 5), 훼2=(3 5), 훼3=(3 4), 훼4=(2 5), 훼5=(2 3), 훼6=(2 4), 훼7=(1 5), 훼8=(1 4), 훼9= (1 3), 훼10= (1 2). The cycles of length 3 are, 휎1=(1 2 3), 휎2=(1 3 2), 휎3=(1 2 4), 휎4=(1 4 2), 휎5=(1 2 5), 휎6=(1 5 2), 휎7=(1 3 4), 휎8=(1 4 3), 휎9=(1 4 5), 휎10=(1 5 4), 휎11=(1 3 5), 휎12=(1 5 3), 휎13=(2 3 4), 휎14=(2 4 3), 휎15= (2 3 5), 휎16=(2 5 3), 휎17= (2 4 5), 휎18= (2 5 4), 휎19= (3 4 5), 휎20= (3 5 4). The cycles of length 4 are, 휏1=(2 3 4 5), 휏2=(2 5 4 3), 휏3=(2 3 5 4), 휏4=(2 4 5 3), 휏5=(2 4 3 5), 휏6=(2 5 3 4), 휏7=(1 2 3 4), 휏8=(1 4 3 2), 휏9=(1 2 3 5), 휏10=(1 5 3 2), 휏11=(1 2 4 3), 휏12=(1 3 4 2), 휏13=(1 2 4 5), 휏14=(1 5 4 2), 휏15=(1 2 5 3), 휏16=(1 3 5 2), 휏17=(1 2 5 4), 휏18=(1 4 5 2), 휏19=(1 3 4 5), 휏20=(1 5 4 3), 휏21=(1 3 5 4), 휏22=(1 4 5 3), 휏23=(1 3 2 4), 휏24=(1 4 2 3), 휏25=(1 3 2 5), 휏26=(1 5 2 3), 휏27=(1 4 3 5), 휏28=(1 5 3 4), 휏29=(1 4 2 5), 휏30=(1 5 2 4).