20 Chapter 1. Groups I A + A = ∅. The reader may verify associativity by showing that both (A + B) + C and A + (B + C) are described by Figure 1.4. Example 1.19. An n × n matrix A with real entries is called nonsingular if it has an inverse that is, there is a matrix B with AB = I = BA, where I is the n×n identity matrix. Since (AB)−1 = B−1A−1, the product of nonsingular matrices is itself nonsingular. The set GL(n, R) of all n × n nonsingular matrices having real entries, with binary operation matrix multiplication, is a (nonabelian) group, called the general linear group. [The proof of associativity is routine, though tedious a “clean” proof of associativity can be given (Corollary 2.128) once the relation between matrices and linear transformations is known.] Definition. If G is a group and a ∈ G, define the powers12 ak, for k ≥ 0, inductively: a0 = 1 and an+1 = aan. If k is a positive integer, define a−k = (a−1)k. A binary operation allows us to multiply two elements at a time how do we multiply three elements? There is a choice. Given the expression 2 × 3 × 4, for example, we can first multiply 2 × 3 = 6 and then multiply 6 × 4 = 24 or, we can first multiply 3 × 4 = 12 and then multiply 2 × 12 = 24 of course, the answers agree, for ordinary multiplication of numbers is associative. If a binary operation is associative, the notation abc is not ambiguous. Let us now consider the definition of powers. The first and second powers are fine: a1 = aa0 = a1 = a and a2 = aa. There are two possible cubes: we have defined a3 = aa2 = a(aa), but there is another reasonable contender: (aa)a = a2a. If we assume associativity, then these are equal: a3 = aa2 = a(aa) = (aa)a = a2a. There are several possible products of a with itself four times is it obvious that a4 = a3a = a2a2? And what about higher powers? Let G be a set with a binary operation an expression in G is an n-tuple (a1,a2,...,an) ∈ G × · · · × G (n factors) which is rewritten as a1a2 · · · an. An expression yields many elements of G by the following procedure. Choose two adjacent a’s, multiply them, and obtain an expression with n − 1 factors: the new product just formed and n − 2 original factors. In this shorter new expression, choose two adjacent factors (either an original pair or an original one together with the new product from the first step) and multiply them. Repeat this procedure until there is a penultimate expression with only two factors multiply them and 12The terminology x square and x cube for x2 and x3 is, of course, geometric in origin. Usage of the word power in this context arises from a mistranslation of the Greek dunamis (from which dynamo derives) used by Euclid. Power was the standard European rendition of dunamis for example, the first English translation of Euclid, in 1570, by H. Billingsley, renders a sentence of Euclid as, “The power of a line is the square of the same line.” However, contemporaries of Euclid (e.g., Aristotle and Plato) often used dunamis to mean amplification, and this seems to be a more appropriate translation, for Euclid was probably thinking of a one-dimensional line segment sweeping out a two-dimensional square. (I thank Donna Shalev for informing me of the classical usage of dunamis.)

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