Answer to Question #194765 in Abstract Algebra for Akmalzhon Nurmakha

Question

Find the cyclic subgroups of U(21).

Answer 1

A subgroup H of a group G is called cyclic if H= <a> , for some a in H

U(n) = { a : a<n and gcd (a, n ) =1}

U(21) = {1, 2, 4, 5 , 8, 10, 11, 13, 16, 17 ,19, 20}

<1>= {1}

<2>={2,4,8,16,11,1}

<4>={4,16,1}

<5>={5,4,20,16,17,1}

<8>={8,1}

<10>={10,16,13,4,19,1}

<11>={11,16,8,4,2,1}

<13>={13,1}

<16>={16,4,1}

<17>={17,16,20,4,5,1}

<19>={19,4,13,16,10,1}

<20>={20,1}

So, there are 3 cyclic subgroups of order 2 : <8> , <13>, <20>

   there is 1 cyclic subgroup of order 3 : <4>

  there are 3 cyclic subgroups of order 3 : <2> , <5>, <10>