Hw8 Problem 4

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Problem 4

Find the index of $\langle 4 \rangle$ in $\mathbb{Z}_{24}$

Solution

By definition of index

(1)

\begin{align} (G:H)= \frac{|G|}{|H|} \end{align}

$\langle 4 \rangle = \{0,4,8,12,16,20\}$ so $|\langle 4 \rangle | = 6$
then,

(2)

\begin{align} (\mathbb{Z}_{24} :\langle 4 \rangle)= \frac{24}{6} = 4 \end{align}