MATH 245 CH9

# MATH 245 CH9 - DISCRETE MATHEMATICS W W L CHEN c W W L Chen...

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DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1991, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 9 GROUPS AND MODULO ARITHMETIC 9.1. Addition Groups of Integers Example 9.1.1. Consider the set Z 5 = { 0 , 1 , 2 , 3 , 4 } , together with addition modulo 5. We have the following addition table: + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 It is easy to see that the following hold: (1) For every x, y Z 5 , we have x + y Z 5 . (2) For every x, y, z Z 5 , we have ( x + y ) + z = x + ( y + z ). (3) For every x Z 5 , we have x + 0 = 0 + x = x . (4) For every x Z 5 , there exists x 0 Z 5 such that x + x 0 = x 0 + x = 0. Definition. A set G , together with a binary operation * , is said to form a group, denoted by ( G, * ), if the following properties are satisfied: (G1) (CLOSURE) For every x, y G , we have x * y G . (G2) (ASSOCIATIVITY) For every x, y, z G , we have ( x * y ) * z = x * ( y * z ). (G3) (IDENTITY) There exists e G such that x * e = e * x = x for every x G . (G4) (INVERSE) For every x G , there exists an element x 0 G such that x * x 0 = x 0 * x = e . Chapter 9 : Groups and Modulo Arithmetic page 1 of 5

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Discrete Mathematics c W W L Chen, 1991, 2008 Here, we are not interested in studying groups in general. Instead, we shall only concentrate on groups that arise from sets of the form Z k = { 0 , 1 , . . . , k - 1 } and their subsets, under addition or multiplication modulo k . It is not difficult to see that for every k N , the set Z k forms a group under addition modulo k . Conditions (G1) and (G2) follow from the corresponding conditions for ordinary addition and results on congruences modulo k . The identity is clearly 0. Furthermore, 0 is its own inverse, while every x 6 = 0 clearly has inverse k - x .
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