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postulate in Axiom 1 is a different property: it says that when we are combining three elements we may group them, two by two, in either of the two obvious ways; the same answer results A group that is commutative is called abelian in honor of Niels Henrik Abel (1802 1829) EXAMPLE 93 Let G be the positive real numbers and let the group operation be multiplication: P(x, y) = x y, where is ordinary multiplication of reals Then (G, P) is a group Axiom 1: Of course multiplication of real numbers is associative Axiom 2: The number 1 is the identity element for multiplication: 1 x = x 1 = x for any real number x Axiom 3: The multiplicative inverse of a group element is its ordinary reciprocal That is, if x R satis es x > 0 then 1/x is its multiplicative inverse EXAMPLE 94 Let G be the integers and let P(x, y) = x + y (ordinary addition) Then (G, P) is a group Axiom 1: Certainly addition of integers is associative Axiom 2: The number 0 is the additive identity Axiom 3: The additive inverse of a group element is its negative: if m Z then m is its group inverse EXAMPLE 95 Let G be the k k matrices with real entries and nonzero determinant This is sometimes called the general linear group on k letters and is denoted by GL(k, R) Let P be ordinary matrix multiplication Then (G, P) is a group Axiom 1: Matrix multiplication is associative Axiom 2: The group identity is the matrix 1 0 0 0 0 1 0 0 k Ik 0 0 1 0 0 0 0 1 k Thus, if m G, then Ik m = m Ik = m
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Axiom 3: The multiplicative inverse of a group element is its matrix inverse Thus if m G then the inverse matrix m 1 is the group inverse Notice in this example that it is important to restrict attention to square matrices, so that multiplication of any two elements in any order will make sense We also require that each matrix have nonzero determinant, so that each matrix will have an inverse To see that G is closed under the group operation of matrix multiplication, we must note that if M, N G then det(M N ) = (det M)(det N ) = 0 Unlike the previous two examples, this last one is a noncommutative group The advantage of the axiomatic method, in the present context, is that when we prove a proposition or theorem about a group G, it applies simultaneously to all groups Thus the axiomatic method gives us both a way of being concise and a way of cutting to the heart of the matter Proposition 91 The multiplicative identity for a group is unique Proof: Then Let G be a group Let e and e both be elements of G that satisfy Axiom 2 e =e e =e Thus e and e must be the same group element Proposition 92 Let G be a group and g G Then there is only one multiplicative inverse for g Proof: Suppose that h and k both satisfy the properties of the multiplicative inverse (Axiom 3) relative to g Then h = h e = h (g k) = (h g) k = e k = k Thus h and k must be the same group element, establishing that the multiplicative inverse is unique Proposition 93 Let g be an element of the group G Then (g 1 ) 1 = g Proof: Observe that g g 1 = e and g 1 g = e
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