Hw8 Problem 6

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Problem 6

Let $|a|=30$ .How many left cosets of $\langle a^4\rangle$ in $\langle a\rangle$ are there?

Solution

$\langle a^4\rangle$ is a subgroup of $\langle a\rangle$. So the order of $\langle a^4\rangle$ is 15 by Theorem 6.14 because $gcd(4,30)=2$ and $30/2=15$.
By the definition of index, we know the number of left cosets of $\langle a^4\rangle$ in $\langle a\rangle$ is $30/15=2$.