Question

The number of generators of Z20
is​

Answer 1

SOLUTION

TO DETERMINE

$\sf{The \: number \: of \: generators \: of \: \: \mathbb{Z}_{20}}$

EVALUATION

$\sf{Let \: \: G = \: \: \mathbb{Z}_{20}}$

Now < k > is a generator iff gcd ( k, 20) = 1

Then k = 1 , 3 , 7 , 9 , 11 , 13 , 19

Now

< 1 > = { 1 , 2 , 3 , 4 , 5 , 6 , 7 ,..., 19 }

< 3 >

= { 3 , 6 , 9 , 12 , 15 , 18 , 2 ,..., 1 , 0}

< 7 >

= { 7 , 14 , 2 , 9 ,.., 0 }

< 9 >

= { 9 , 18 , 7 , 14 ,... 0 }

< 11 > = < 9 >

< 13 > = < 7 >

< 19 > = < 1 >

So the generators are

< 1 > , < 3 > , < 7 > , < 9 >

Hence the number of generators = 4

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