Let m and n be integers. By mZ+nZ we mean the set {a + b a € mZ,b ≤ nZ} of all sums of anelement of mZ and nZ, i.e., a typical element of the group looks like xm + yn for some integers x andy.(a) Show that mZ+nZ is a subgroup of Z.(b) Find a generator of the subgroup 12Z +21Z.

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Answered: Integers. set {a and nZ, i.e., a…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 21E: Exercises 21. Suppose is a cyclic group of order. Determine the number of generators of for each...

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Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an isomorphism from G to G that is different from the one given in Example 5 of this section. Example 5 Consider G=1,i,1,i under multiplication and G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. In order to define a mapping :G4 that is an isomorphism, one requirement is that must map the identity element 1 of G to the identity element [ 0 ] of 4 (part a of Theorem 3.30). Thus (1)=[ 0 ]. Another requirement is that inverses must map onto inverses (part b of Theorem 3.30). That is, if we take (i)=[ 1 ] then (i1)=((i))1=[ 1 ] Or (i)=[ 3 ] The remaining elements 1 in G and [ 2 ] in 4 are their own inverses, so we take (1)=[ 2 ]. Thus the mapping :G4 defined by (1)=[ 0 ], (i)=[ 1 ], (1)=[ 2 ], (i)=[ 3 ]
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Integers. set {a and nZ, i.e., a typical element of the group looks like xm +yn for some integers and mZ+nZ is a subgroup of Z. erator of the subgroup 12Z + 21Z.