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Example 4.13 the order of z n is n 2 example 4.14 the

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Unformatted text preview: Example 4.13 The order of Z n is n . 2 Example 4.14 The order of Z * p for prime p is p- 1. 2 Note that in specifying a group, one must specify both the underlying set G as well as the binary operation; however, in practice, the binary operation is often implicit from context, and by abuse of notation, one often refers to G itself as the group. Usually, instead of using a special symbol like ? for an abelian group operator, one instead uses the usual addition (“+”) or multiplication (“ · ”) operators. If an abelian group G is written additively, then the identity element is denoted by 0 G (or just 0 if G is clear from context), and the inverse of an element a ∈ G is denoted by- a . For a,b ∈ G , a- b denotes a + (- b ). If n is a positive integer, then n · a denotes a + a + ··· + a , where there are n terms in the sum. Moreover, if n = 0, then n · a denotes 0, and if n is a negative integer then n · a denotes- ((- n ) · a ). If an abelian group G is written multiplicatively, then the identity element is denoted by 1 G (or just 1 if G is clear from context), and the inverse of an element a ∈ G is denoted by a- 1 or 1 /a . As usual, one may write ab in place of a · b . For a,b ∈ G , a/b denotes a · b- 1 . If n is a positive integer, then a n denotes a · a · ··· · a , where there are n terms in the product. Moreover, if n = 0, then a n denotes 1, and if n is a negative integer, then a n denotes ( a- n )- 1 . For any particular, concrete abelian group, the most natural choice of notation is clear; however, for a “generic” group, the choice is largely a matter of taste. By convention, whenever we consider a “generic” abelian group, we shall use additive notation for the group operation, unless otherwise specified. We now record a few simple but useful properties of abelian groups. Theorem 4.3 Let G be an abelian group. Then 1. for all a,b,c ∈ G , if a + b = a + c , then b = c ; 2. for all a,b ∈ G , the equation a + x = b in x has a unique solution in G ; 3. for all a,b ∈ G ,- ( a + b ) = (- a ) + (- b ) ; 4. for all a ∈ G ,- (- a ) = a ; 5. for all a ∈ G and all n ∈ Z , (- n ) a =- ( na ) = n (- a ) . Proof. Exercise. 2 If G 1 ,...,G k are abelian groups, we can form the direct product G 1 ×···× G k , which consists of all k-tuples ( a 1 ,...,a k ) for a 1 ∈ G 1 ,...,a k ∈ G k . We can view G 1 × ··· × G k in a natural way as an abelian group if we define the group operation “component wise”: ( a 1 ,...,a k ) + ( b 1 ,...,b k ) := ( a 1 + b 1 ,...,a k + b k ) . 21 Of course, the groups G 1 ,...,G k may be different, and the group operation applied in the i th component corresponds to the group operation associated with G i . We leave it to the reader to verify that G 1 × ··· × G k is in fact an abelian group....
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Example 4.13 The order of Z n is n 2 Example 4.14 The order...

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