Group Theory: Basic Concepts

⋆ Group: (G, γ) where G is a set and γ a binary operation γ : G×G → G. If γ is clear from the context the group is simply denoted by G. ◦ In place of γ(f, g) write fγg or simply fg when there is no ambiguity, as in most of what follows. ◦ Example: G = Z = {. . .− 2,−1, 0, 1, 2 . . .}, γ = +, γ(f, g) = f + g. • For G to be a group three conditions must be satisfied: (i) γ is associative: γ(f, γ(g, h)) = γ(γ(f, g), h), or f(gh) = (fg)h. (ii) There is an identity e ∈ G: eg = ge = e for all g ∈ G. (iii) Every g in G has an inverse ḡ, usually denoted by g, such that gḡ = ḡg = e.