Question

Q. Find index of subgroup generated by 3 in Z12.

Answer 1

Answer on Question #58978 – Math – Abstract Algebra

Question

Find index of subgroup generated by 3 in $Z_{12}$.

Solution

$\mathrm{H} = \mathrm{h}_{3\mathrm{i}}$ and $\mathrm{G} = \mathrm{Z}_{12}$. Note that $\mathrm{H} = \mathrm{h}_{3\mathrm{i}} = \{0, 3, 6, 9\}$, therefore, $|\mathrm{H}| = 4$. Since $|\mathrm{G}| = |\mathrm{Z}_{12}| = 12$, by Lagrange’s Theorem, the index of subgroup generated by 3 in $\mathrm{Z}_{12}$ is $[\mathrm{G} : \mathrm{H}] = |\mathrm{G}| / |\mathrm{H}| = 12/4 = 3$.

Answer: 3.

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