Introductory Notions

Abstract

Definition 1.1. A group G is a non empty set in which it is defined a binary operation, i.e. a function: (Formula presented.) such that, if ab denotes the image of the pair (a, b), i)the operation is associative: (ab)c = a(bc), for all triples of elements a, b, c ∈ G;ii)there exists an element e ∈ G such that ea = a = ae, for all a ∈ G. This element is unique: if e′ is also such that e′a = a = ae′, for all a ∈ G, ea = a implies, with a = e′, that ee′ = e′, and a = ae′ implies, with a = e, that ee′ = e. Thus e′ = e;iii)for all a ∈ G, there exists b ∈ G such that ab = e = ba.