geduiwelh

2021-01-31

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

hesgidiauE

Solution: Given group $G=\left({Z}_{15},\oplus \right)$ a) We know that by Lagranges theorem order of subgroup divide the order of group. Since $O\left(G\right)=15$ Let H be subgroup of G. Then $O\left(H\right)|O\left(G\right)\cdot O\left(H\right)|15$ Possible order of H are 1, 3, 5, 15 Since $G-\left({Z}_{15},\oplus \right)$ 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 ${H}_{1},{H}_{2},{H}_{3}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{H}_{4}$ are subgroup of group $G=\left({Z}_{15},\oplus \right)$

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