Relevant Topics from Abstract Algebra

Abstract

AbstractIn 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.