Question

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

Polynomial arithmetic

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

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