Show that G = {1, -1} is a group under multiplication.

We must prove the four axioms of a group are satisfied (i.e Identity, Associativity, Closure, and Invertibility). Identity: The identity element is an element e in G such that ex = xe = x for all x in G. Clearly in this instance e = 1 since 11 = 1 and 1(-1) = -1. Associativity: A group is associative if for all a,b,c in G, a*(bc) = (ab)c. In this instance the associativity clearly follows from the associativity of multiplication. Closure: A group is closed if the product of any two elements of the group is also a member of the group. In this example there are four possible products of the elements, namely 11, 1*(-1), (-1)1, and 11. All of these are either 1 or -1, hence all the products are members of the group {1, -1} so G is closed. Invertibility: To show invertibility we must show that every element of G has an inverse, i.e. for all x in G there exists x^-1 in G such that xx^-1 = x^-1x = e, where e is the identity element (in this case e = 1). Clearly 11 = 1 so that 1^-1 = 1. For -1 we see that (-1)(-1) = 1, hence -1^-1 = -1. So both 1 and -1 have inverses which are in G, thus G is closed. So all the axioms are satisfied, therefore G is a group.