- #1

Benzoate

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## Homework Statement

Find all generators of Z(6), Z(8) , and Z(20)

## Homework Equations

## The Attempt at a Solution

I should probably list the elements of Z(6), Z(8) and Z(20) first.

Z(6)={0,1,2,3,4,5}

Z{8}={0,1,2,3,4,5,6,7}

Z(20)={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}

perhaps taking the divisors of Z(sub 6) , Z(sub 8) and Z(sub 20)

For 6 , the divisors are 1,2,3,and 6

For 8 , the divisors are 1,2,4,and 8

For 20 , the divisors are 1,2,4,5, and 20

any element in <..> represent the generater

Z(6), its generators are <a^1> , <a^2> , <a^3> and <a^6>

<a^1> ={0,1,2,3,4,5}

<a^2>={0,2,4}

<a^3>={0,3}

and <a^6>={0}

<a^1> has order 6, <a^2> has order 3 ,<a^3> has order 2 and <a^6> has order 1.

the back of my book says the generators for generator 6 are 1 and 5. I don't understand why generator 6 is 1 and 5.