THE MULTIPLICATIVE GROUP OF NON-ZERO COMPLEX NUMBERS

May 11th, 2015
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We will call a group that consists of a single element x ∈ G (called the group generator) a cyclic group generated by x and denoted by < x > if written in multiplication notation as G = < x > = {1, x, x 2 ,..........., x n −1} where x n = 1 . In particular, observe that the exponents of x include all negative integer as well as 0 and the positive integers ( x 0 is defined to be the identity). A finite *

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RuppROYAL UNIVERSITY OF PHNOM PENHTHE MULTIPLICATIVE GROUP OF NON-ZERO COMPLEX NUMBERS IS ISOMORPHIC TO THE MULTIPLICATIVE GROUP OF THE UNIT CIRCLEA Thesis In Partial Fulfillment of the Requirement for the Degree of Master of Science (Mathematics)We will call a group that consists of a single element x G (called the group generator) a cyclic group generated by x and denoted by < x > if written in multiplication notation as G = < x > = {1, x, x 2 ,..........., x n 1} where x n = 1 . In particular, observe that the exponents of x include all negative integer as well as 0 and the positive integers ( x 0 is defined to be the identity). A finite *, thecyclic group generated by x is necessarily abelian, and can be written in addition notation as G = < x > = { nx : n } = {0, x, 2 x,........., (n 1) x } . Clearly if there is n such that nx = 0 , we say that < x > is finite. Otherwise x is infinite. Similarly, if G is an infinite cyclic group generated by x , then G must be abelian and can be written as{1, x, x 2 ,...............} or in addition notation {0, x, 2 x,.................} . In this case, G is isomorphic to additive groupof all integers.ITH PHANNY9Royal University of Phnom PenhMaster of Science in Mathematics IntroductionExample 18: The ( , +) is a cyclic group generated by 1 or 1 ; that is,= < 1 > = { n(1) : n } and= < 1 > = { n(1) : n } .Example 19: The set ({1, 1 }, ) is a cyclic multiplicative gr

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