Solution: Given group \(\displaystyle{G}={\left({Z}_{{15}},\oplus\right)}\) a) We know that by Lagrange`s theorem order of subgroup divide the order of group. Since \(\displaystyle{O}{\left({G}\right)}={15}\) Let H be subgroup of G. Then \(\displaystyle{O}{\left({H}\right)}{\left|{O}{\left({G}\right)}\cdot{O}{\left({H}\right)}\right|}{15}\) Possible order of H are 1, 3, 5, 15 Since \(\displaystyle{G}-{\left({Z}_{{15}},\oplus\right)}\) is cycling group. Then every divisor of order of group has subgroup. Then b)

\(H_1 = \left\{ e \right\}\)

\(H_2 = <5> = \left\{ 5, 10, 0 \right\}\)

\(H_3 = <5> \left\{ 3, 6, 9, 12, 0 \right\}\)

\(H_4 = <1> = \left\{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0 \right\}\)

These are \(\displaystyle{H}_{{1}},{H}_{{2}},{H}_{{3}}{\quad\text{and}\quad}{H}_{{4}}\) are subgroup of group \(\displaystyle{G}={\left({Z}_{{15}},\oplus\right)}\)