Abstract Algebra The word „Algebra‟ is derived from the Arabic word „al-jabr‟. Classically, algebra involves the study of equations and a number of problems that devoted out of the theory of equations. Then the term modern algebra is used as a tool to describe the information based on detailed investigations. Consequently a special branch of algebra namely abstract algebra, a generalization of modern algebra is introduced in which the algebraic systems (structures) are defined through axioms. The present course is divided into three blocks. The first block containing two units deal with the advanced study of group and ring theory. Then second block containing two units, introduces the concepts of vector spaces and linearly independence, basis and dimensions including the theory of linear transformations in consecutive units 3 and 4. Finally in block III the interesting properties of a special class of vector spaces called inner product spaces have been established and it is unit 5. After the study of this course the students will realize that abstract algebra allow us to deal with several simple algebraic systems by dealing with one representative system. Block Introduction In previous classes the students have already studied elementary abstract algebra in which they have grabbed the elementary knowledge of two algebraic systems namely groups and rings including their properties. The main aim of this block is to deal with the further studies of these two systems. Ideally the goal is to achieve the information regarding groups and rings concerned with the prescribed course. Units – 1This unit provides the special case of isomorphism of groups called automorphism and inner automorphism. Consequently automorphism group is described. Thereafter we also introduced conjugacy relation, class equation and counting of the conjugate elements corresponding to the elements of group. In the end of this unit, we shall discuss that a finite group G has a subgroup of every prime order dividing the order of group through Cauchy‟s & Sylow‟s theorems. Unit – 2This unit presents the study of ring homomorphism and ideal (analogue of normal subgroup), quotient ring (analogue of quotient group), field of quotient group of an integral domain and a special class of integral domain called Euclidean ring. Then we introduced the polynomial rings over rational field and commutative rings. In the end of the unit, unique factorization domain has been discussed. Some self study examples and proof of the theorems are left to the readers to check their progress. 2 BLOCK – 1 ABSTRACT ALGEBRA UNIT – 1 GROUP THEORY Structure : 1.1Automorphism 1.2Inner Automorphism 1.3Automorphism groups 1.4Conjugacy relation and Centralizer 1.5Normaliser 1.6Counting principle and the class equation of a finite group 1.7 Cauchy’s and Sylow’s Theorems for finite abelian and non abelian groups. *Points for Discussion/ Clarification **References Before describing the main contents of unit, first we shall explain the fundaments concepts of group theory which are essential to understand the entire unit for convenience. Students may wish to review this quickly at first and then read the further required part of the prescribed course. 1.Function : The Notation f : A B to denote a function (or map) from A to B and the value of f at a is denoted by f(a). The set A is called the domain of f and B is called the codomain of f. The set f (A) = {bB| b = f(a),for some aA} is called the range or image of A under f. For each subset C of B, the set f –1 (C) = { aA: f(a) C} Consisting of the elements of A mapping into C under f is called the pre image or inverse image of C under f. Let f : AB. Then (a) f is injective (or injection or one one) if whenever 3 a1 a2, then f(a1) f(a2) OR f (a1) =f (a2), then a1 = a2 (b) f is surjective (or surjection or onto) if for each bB there is some aA such each that f(a) = b. (c) f is bijective (or bijection) if f is both injective and surjective 2. Permutation : A permutation on a set A is simply a bijection from A to it self. 3. Binary relation : A binary relation “” on a set A is a subset R of A A and we write a b if (a, b) R . The relation “” on A is a) reflexive if a a , for all a A b) symmictric if a b b a , for all a, b A c) transitive if a b and b c a c , for all a, b, c A A relation is an equivalence if it is reflexive, symmetric and transitive. 4.Equivalence Class : If “ ” defines an equivalence relation on A, then equivalence class of a A is defined to be {x A| x a}. It is denoted by the symbol [a]. i.e. [a] = {x A| x a}. Elements of [a] are said to be equivalent to a. 5.Binary operation or composition : Let G be a non empty set. Then a binary operation on G is a mapping from G G in to G, defined by (a, b) ab i.e. a, b G ab G Example : Let A = { 1, –1} & B = {1, 2} Then the multiplication operation „.‟ is a binary operation on A but not B. Since for 2, 2 B 2.2 = 4 B. 6. Group: A non empty set G along with above binary operation is called a group if it satisfies the following axioms : G1 For a, b G a b G (Closure axiom) G2 For a, b, c G (a b) c = a (b c) (associative axiom) G3 There exists an element e G called an identity element of G such that for all a G a e = a = e a –1 –1 –1 G4 For every a G , there exist a G. Such that aa = e = a a . –1 a is called the inverse of a in G. The axiom G1 is a super flous statement, i.e. It is a binary composition. G with axiom G1 only is called a groupoid, with G1 andG2 is called semi group and with G1, G2 and G3 only is called a monoid. 4 7.Commutative or abelian group : A group G is called an abelian or commutative group if for all a, b G ab = ba A group G is called finite if it is a finite set , other wise infinite. 8. Order of Group : The number of elements in a group G is called the order of G and It is denoted by 0(G). The infinite group is said to be of infinite order. The smallest group is denoted by {c} consisting only the identity element. It is clear that 0 ({e}) = 1. Example : (i) The set Z (or I) of integers forms an abelian an group w.r.t usual addition of integers but Z does not form a group w.r.t. multiplication since multiplicative inverse of every element of Z does not belong to Z. For example 2 Z but 2–1 = ½ Z / (ii) The set Q of rationals , R the set of reals are abelian groups w.r.t addition. (iii) Set of all 2 2 matrices over integers under matrix addition forms an abelian group. 9. Properties of group : In a group G : (i) The identity element is unique. (ii) The inverse of each a G is unique. (iii)(a–1)–1 = a , for all a G (iv) (ab)–1 = b–1 a–1 for all a, b G (Reversal law) (v) Left cancellation law ab = ac b = c (vi) Right cancellation law ba = ca b = c for all a, b, c G 10. Complex of a group : Every nonempty subset of a group G is called a Complex of G. 11. Sub Group : A non empty sub set H of a group G is a sub group of G if H is closedunder products and inverses . i.e. a, b H ab H and a–1 H The sub groups H = G and {e} are called trivial or improper sub groups of G and the sub groups H G and {e} are called nontrivial or proper sub groups of G. It can be easily seen that the identity of a sub group H is the same as the identity of group and the inverse of a in H is the same as the inverse of a in G. Example : The set {1, –1} is a sub group of the multiplicative group {1, –1, i, –i} (ii) The set of even integers {0, 2, 4, …} is a subgroup of the set of additive group Z = { 0, 1, 2, ….} of integers. Criterian for a sub group : A sub set H of a group G is a sub group if and only if (i) H and (ii) For all a, b H a.b H and for all a H a–1 H 5 OR For all a, b H ab–1 H It can also be check that the intersection of two subgroups is a subgroup but union of two subgroups is not necessarily a subgroup. 12 Order of an Element : The order of an element a G is the least positive integers n such that an = e , where e is the identity of G. Example : Let G = {1, –1. i, –i}. Then G is a multiplicative group. Now 11 = 1 0(1) = 1 (–1)1 = 1 , (–1)2 = 1 0 (–1) = 2, i1 = I, i2 = –1, i3 = –i, i4 = 1 o(i) = 4 (–i)1 = –i, (–i)2 = –1, (–i)3 = i, (–i)4 = 1 0(–i) = 4 13 Cyclic group : If in a group G there exist an element a G such that every element x G is of the form am, where m is some integer.