Discrete Mathematics..

More about Groups 375listed above areandn(−x) =−(nx) = (−n)x for all n ∈ mx + nx = (m + n)x = nx + mx for all m, n ∈ n(mx) = (nm)x = m(nx) for all m, n ∈ n(x + y) = nx + nyfor all n ∈ since additionis commutative.We shall denote a group by (G, ∗) rather than (S, ∗) in order to emphasize thatwe are referring to a group rather than some other algebraic structure.Perhaps the most obvious characteristic of any group (G, ∗) is its ‘size’, that isthe number of elements in the underlying set G. This is termed the ‘order’ of thegroup (G, ∗).Definition 8.9The order of a group (G, ∗) is the cardinality of the set G. It is denoted by|G| (see definition 3.1).We now prove some useful theorems about the properties of groups.Theorem 8.3If (G, ∗) is a group, then the left and right cancellation laws hold; that is, ifa, x, y ∈ G, then(a)(b)ax = ay implies that x = y (left cancellation law), andxa = ya implies that x = y (right cancellation law).ProofSuppose that ax = ay.Since (G, ∗) is a group, then the element a has an inverse a −1 . ‘Multiplying’ on