Question

Consider the "clock arithmetic" group (Z_15, oplus)a) Using Lagrange`s Theotem, state all possible orders for subgroups of this group.b) List all of the subgroups of (Z_15, oplus)

Polynomial arithmetic
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asked 2021-01-31

Consider the "clock arithmetic" group \((Z_{15}, \oplus)\) a) Using Lagrange`s Theotem, state all possible orders for subgroups of this group. b) List all of the subgroups of \((Z_{15}, \oplus)\)

Answers (1)

2021-02-01

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 \(\displaystyle{G}={\left({Z}_{{15}},\oplus\right)}\)

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