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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.

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  • $\begingroup$ Instead of "So all groups of order $p^3$ are not abelian." you surely meant to write "So not all groups of order $p^3$ are abelian." (and - to be picky - the implication "non-abelian of order $p^3$ implies center of order $p$" does not imply the existence of a non-abelian group of order $p^3$). In mathematics preciseness is important. $\endgroup$ – j.p. Mar 24 '15 at 10:23
  • $\begingroup$ ohh so sorry I did a mistake. Yah it will be "not all". Thanks..@j.p. 2 $\endgroup$ – adember Mar 27 '15 at 6:11
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Hint $1$:What about quaternion group of order 8?

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

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