TITLE: $\mathbb Z^n/\langle (a,...,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$
QUESTION [9 upvotes]: I am trying to show the isomorphism
$$\mathbb Z^n/\langle (a,...,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle.$$
I've tried to define $\psi:\mathbb Z^n \to \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$ an epimorphism with $\ker(\psi)=\langle (a,...,a) \rangle$ so to apply the first isomorphism theorem, but I couldn't come up with an appropiate morphism. Any hints or suggestions would be greatly appreciated.

REPLY [9 votes]: If you want to use the first isomorphism theorem, then define $\psi:\mathbb Z^n \to \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$ by $\psi(x_1,\dots,x_n)=(x_2-x_1,\dots,x_n-x_1,\bar x_1)$. Check that $\psi$ is a surjective homomorphism and $\ker\psi=\langle (a,\dots,a) \rangle$.