Consider the "clock arithmetic" group (Z15,⊕) a) Using Lagrange`s Theotem, state all possible orders for...

Solution: Given group ${\displaystyle G=(Z_{15},\oplus )}$ 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 ${\displaystyle O\left(H\right)|O\left(G\right)\cdot O\left(H\right)|15}$ Possible order of H are 1, 3, 5, 15 Since ${\displaystyle G-(Z_{15},\oplus )}$ is cycling group. Then every divisor of order of group has subgroup. Then b)

$H_1=\left\{e\right\}$

$H_2=<5>=\{5,10,0\}$

$H_3=<5>\{3,6,9,12,0\}$

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

These are ${\displaystyle H_1,H_2,H_3\text{and}{H}_4}$ are subgroup of group ${\displaystyle G=(Z_{15},\oplus )}$