list the elements of the subgroup in the group mathbbZ16 of

list the elements of the subgroup in the group mathbb{Z}_{16}, of the integers modulo 16 (under addtion modulo 16).

Solution

Z16 is a cyclic group, Z16 has the divisors equal to {1,2,4,8}

Hence there are four sub-groups of Z16

<0> is the sub-group of order 1

<1> is the sub-group of order 16 since it contains all the elements

<2> = {0,2,4,6,8,10,12,14} sub-group of order 8

<4> = {0,4,8,12} sub-group of order 4

<8> = {0,8} subgroup of order 2.

Hence there will be only one sub-group <4> that will be having 4 elements in the sub-group other sub-groups contains 1,2,8,16 elements respectively