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.

  • $\begingroup$ 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

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

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.