Proposition 19

*Let a be an integer > 0. Let H be a subgroup of $\mathbf{Z}/a\mathbf{Z}$, b the order of H and c its index in $\mathbf{Z}/a\mathbf{Z}$. Then $a = bc$, $H = c\mathbf{Z}/a\mathbf{Z}$ and H is isomorphic to $\mathbf{Z}/b\mathbf{Z}$.

Conversely, let b and c be two integers > 0 such that $a = bc$. Then $a\mathbf{Z} \subset c\mathbf{Z}$ and $c\mathbf{Z}/a\mathbf{Z}$ is a subgroup of $\mathbf{Z}/a\mathbf{Z}$, of order b and index c.

$a = bc$ by no. 4, Corollary to Proposition 4. By no. 7, Theorem 4, H is of the form $H'/a\mathbf{Z}$, where $H'$ is a subgroup of $\mathbf{Z}$ and $\mathbf{Z}/H'$ is isomorphic to $(\mathbf{Z}/a\mathbf{Z})/H$ and hence of order c. By Proposition 17 and the Corollary to Proposition 16, $H' = c\mathbf{Z}$ and hence H is monogenous. Finally, H is isomorphic to $\mathbf{Z}/b\mathbf{Z}$ by Proposition 18. Conversely, if $a = bc$, then $a\mathbf{Z} \subset c\mathbf{Z}$ for $a \in c\mathbf{Z}$: the quotient group $(\mathbf{Z}/a\mathbf{Z})/(c\mathbf{Z}/a\mathbf{Z})$ is isomorphic to $\mathbf{Z}/c\mathbf{Z}$ (no. 7, Theorem 4) and hence of order c (no. 4, Corollary to Proposition 4) and index b (no. 4, Corollary to Proposition 4).