Relevant Topics from Abstract Algebra

Abstract

In this chapter, we discuss the relevant topics from abstract algebra which are later used extensively. We start with the definition of groups. A set $\mathbb {G}$ with some binary operation o over $\mathbb {G}$, is called a group if all the following properties (also known as group-axioms) are satisfied: Closure $(G_1)$: for every $a, b \in \mathbb {G}$, the element $a \;o\; b \in \mathbb {G}$. Associativity $(G_2)$: for every a, b, c, the condition $(a \;o\; b) \;o\; c = a \;o\; (b \;o\; c)$ holds. Existence of identity $(G_3)$: there exists a unique identity element $e \in \mathbb {G}$, such that for every $a \in \mathbb {G}$, the condition $(a \;o\; e) = (e \;o\; a) = a$ holds. Existence of inverse $(G_4)$: for every $a \in \mathbb {G}$, there exists a unique element, say $a^{-1} \in \mathbb {G}$, such that the condition $(a \;o\; a^{-1}) = (a^{-1} \;o\; a) = e$ holds. We note that in Definition 2.1, the operation o need not satisfy the commutative property. If apart from the axioms $G_1, \ldots , G_4$, the operation o satisfies the commutative property, then $\mathbb {G}$ along with the operation o is called as an Abelian group.