which of the following options are true?

Question

Let $G$ be a group, which are true?

1. $G$ has a nontrivial centre $C$, then $G/C$ has trivial centre.

2. If $G \not = 1$, there exists a nontrivial homomorphism $h: \Bbb Z\to G$.

3. If $|G|=p^3$, for $p$ is a prime, then $G$ is an abelian group.

4. If $G$ is a nonabelian group, then it has a nontrivial automorphism.

My proceed: For option 3 we know that a noncommutative group of order $p^3$ has the centre of order $p$. So all groups of order $p^3$ are not abelian. So option 3 is false. For option 2 there always exists a nontrivial homomorphism from $\mathbb Z$ to $\mathbb Z_n$. So option 2 is true, but I cannot prove/ disprove other two options. Please someone help.

Thanks in advance.

Answer 1

Hint $1$:What about quaternion group of order 8?

$4$.What happens if all the automorphism(In particular inner automorphisms) are trivial?