Abstract Algebra: Theory and Applications · Problem 10.4.8

The key insight is that generators survive the quotient map: if $G = \langle a \rangle$, then every element of $G/H$ can be written as a power of the coset $aH$. Since the quotient group inherits the cyclic structure from $G$ through this coset, $G/H$ is generated by a single element, making it cyclic. Use the fact that any $gH \in G/H$ can be expressed as $(aH)^k$ for some integer $k$.

The key insight is that generators survive the quotient map: if $G = \langle a \rangle$, then every element of $G/H$ can be written as a power of the coset $aH$. Since the quotient group inherits the cyclic structure from $G$ through this coset, $G/H$ is generated by a single element, making it cyclic. Use the fact that any $gH \in G/H$ can be expressed as $(aH)^k$ for some integer $k$.