Groups I

A group is an ordered pair, (G, ∗), consisting of a set G and a binary operation ∗: G × G → G (( x, y ) ↦ x ∗ y ) satisfying (a) ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ) for all x, y, z ∈ G (associativity), (b) there exists an element e ∈ G such that x ∗ e = x = e ∗ x for all x ∈ G , (c) for each x ∈ G , there exists an element x ′ ∈ G such that x ∗ x ′ = e = x ′ ∗ x . The operation ∗ is frequently written as a product, i.e. ( x, y ) ↦ xy , or as a sum, ( x, y ) ↦ x + y . We then say that ( G , ∗) is respectively a multiplicative or an additive group. The element e , which can be shown to be unique, is called the identity or neutral element of G . When G is multiplicative it is denoted by 1 and when G is additive by o. Correspondingly, x ′, the inverse element of x (again unique) is denoted by x –1 and – x respectively. (Cf. p. 15.) Note that the definition of a group given above is wasteful in the sense that not all the conditions listed are independent (p. 6). Thus, for example, one can omit the condition x = e ∗ x from ( b ) and the condition e = x ′ ∗ x from ( c ), since these relations can be deduced from the remaining conditions.