Find all the quotient groups of integers Z /n?

Question

I have tried it and I think if Z is cyclic group then all of its quotient groups are Cyclic.. But what type of normal subgroups of Z/n. Please help me to get a better idea of what I need...

Answer 1

You're right that the quotients of a cyclic group are always cyclic.

Since $\mathbb Z/n$ is cyclic, it is abelian, and so every subgroup is normal.

To finish, you need to find all subgroups of $\mathbb Z/n$ and the corresponding quotients. It's not hard.

Answer 2

In a finite cyclic group, there's a unique (normal) subgroup of every order dividing the order of the group.

Every quotient of $\Bbb Z_n$ is a homomorphic image of $\Bbb Z_n$ ( use the canonical projection), hence cyclic.

In conclusion, you get a cyclic subgroup of every order dividing the order of the group.

If you're talking about $\Bbb Z$ (I'm not really sure if you are or not), it can be shown that all subgroups are of the form $n\Bbb Z$. Thus you get as quotients all the finite cyclic groups.