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MTS 211

ABSTRACT ALGEBRA

LECTURE NOTE 2 FOR 2011/2012 FIRST SEMESTER 1.0 ALGEBRAIC STRUCTURE ∗ → . That is ∗ =

Let A be a non-empty set, a binary operation on A is a function ∗ such that ,∗ is A rule by which every pair of elements . Such a set is said to be closed under ∗. EXAMPLE 1.1.0 ,

yield a third element z in A, viz

The usual arithmetic operations +, - ,x ,÷ are binary operations on the Real set. Similarly, the operations ∪,∩, ∆ are binary operations on the power set P(A) . By an algebraic structure (or algebraic system ) we mean a non-empty set S, equipped with one or more binary operations. We denote an algebraic structure consisting of set S and a binary operation ∗ by the ordered pair (S, ∗). Similarly, an algebraic system consisting of set S and two operations ∗ and o shall be denoted by the ordered triple (S, ∗, o ). EXAMPLE: 1.1.1 (ℕ, +), ( , +), (ℚ, .), (ℝ, +, .) (ℂ, +, .) (p(X), ∪) and (P (x), ∪, ∩ ) are all algebraic systems. For any binary operation * defined on a set S, 1. 2. 3. If x * y = y * x for all x,y ∈ X, then * is said to be communicative. If x * (y*z) = (x*y)*z for all x,y,z ∈ S then * is said to be associative. If there is an element e ∈ S such that e*x = x*e = x for all x ∈ S then e is called the

identity element (or unity element) of S. In particular e*e = e. e.g. 0 and 1 are the identity elements of IR with respect to + and . operations respectively since for any x ∈ IR, x + 0 = 0 + x = x, x.1 = 1.x = x. 4. if there is an element y ∈ S such that x*y = y*x = e for x ∈ S, then y is called the inverse

of x in S w.r.t*, where e is the identity element of S.

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If * and o are operations defined on S we say that o is left distributive over * if x o (y*z) = x o y * x o z for all x,y,z ∈ S and o is right distributive over * if (x*y) o z = x o z * y o z for all x,y,z ∈ S.

If o is both left and right distributive over * we simply say o is distributive over *.

1.2

THE STRUCTURE OF GROUPS

1.2.1 DEFINITION AND EXAMPLES OF GROUPS An...