For the Course ” Introduction to Representation Theory ” , Fall 2015

Definition 1.1. A group G is a set with a binary operation G × G → G, called multiplication, such that (1) ∀f, g, h ∈ G.(fg)h = f(gh) (2) ∃1 ∈ Gs.t.∀g ∈ G, 1g = g1 = g (3) ∀g ∈ G, ∃g−1 ∈ Gs.t. gg−1 = g−1g = 1 A morphism of groups φ : G → H is a function φ : G → H s.t. φ(g1g2) = φ(g1)φ(g2)∀g1, g2 ∈ G. Example 1.2. Z the group of integers, Z/nZ = the cyclic group of order n, Sym(X)the group of all bijections from X to itself. Also denoted by Symn or Sn if X has n elements. If V is a vector space of dimension n over a field F then we denote by GL(V ) or by GL(n, F ) the group of all invertible linear transformations from V to itself.