Does this group theory question require an additional hypothesis?












1














The problem is to




Show that if $G$ is a finite group and for all nontrivial elements $a, b$ there exists an automorphism taking $a$ to $b$, then $G$ is a $C_p$ vector space, where $C_p$ is the group of prime order $p$.







My question is if an additional hypothesis that $G$ is abelian is needed.




I cannot seem to prove that $G$ is abelian from the hypotheses.



Of course, the point is that all the elements of $G$ have the same prime order $p$.



But I cannot seem to get the result without showing $G$ is abelian, in which case the normality of all subgroups gives what I want.










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  • If the additional hypothesis is not necessary, then please don't answer the question, by the way.
    – user187877
    Oct 28 '14 at 1:43










  • Are you sure you can't show that the group is already abelian by your hypothesis?
    – Jonny
    Oct 28 '14 at 1:46










  • Can you? I gave it a try but not too hard of a try.
    – user187877
    Oct 28 '14 at 1:48










  • I just saw this result online in the case of abelian groups. But my professor phrased it this way, and I thought he was just trying to be sneaky about saying every element has the same order. But now I'm not so sure.
    – user187877
    Oct 28 '14 at 1:49






  • 2




    These hypotheses can show that $G$ is abelian, so this solves the problem.
    – user187877
    Oct 28 '14 at 2:24
















1














The problem is to




Show that if $G$ is a finite group and for all nontrivial elements $a, b$ there exists an automorphism taking $a$ to $b$, then $G$ is a $C_p$ vector space, where $C_p$ is the group of prime order $p$.







My question is if an additional hypothesis that $G$ is abelian is needed.




I cannot seem to prove that $G$ is abelian from the hypotheses.



Of course, the point is that all the elements of $G$ have the same prime order $p$.



But I cannot seem to get the result without showing $G$ is abelian, in which case the normality of all subgroups gives what I want.










share|cite|improve this question
























  • If the additional hypothesis is not necessary, then please don't answer the question, by the way.
    – user187877
    Oct 28 '14 at 1:43










  • Are you sure you can't show that the group is already abelian by your hypothesis?
    – Jonny
    Oct 28 '14 at 1:46










  • Can you? I gave it a try but not too hard of a try.
    – user187877
    Oct 28 '14 at 1:48










  • I just saw this result online in the case of abelian groups. But my professor phrased it this way, and I thought he was just trying to be sneaky about saying every element has the same order. But now I'm not so sure.
    – user187877
    Oct 28 '14 at 1:49






  • 2




    These hypotheses can show that $G$ is abelian, so this solves the problem.
    – user187877
    Oct 28 '14 at 2:24














1












1








1







The problem is to




Show that if $G$ is a finite group and for all nontrivial elements $a, b$ there exists an automorphism taking $a$ to $b$, then $G$ is a $C_p$ vector space, where $C_p$ is the group of prime order $p$.







My question is if an additional hypothesis that $G$ is abelian is needed.




I cannot seem to prove that $G$ is abelian from the hypotheses.



Of course, the point is that all the elements of $G$ have the same prime order $p$.



But I cannot seem to get the result without showing $G$ is abelian, in which case the normality of all subgroups gives what I want.










share|cite|improve this question















The problem is to




Show that if $G$ is a finite group and for all nontrivial elements $a, b$ there exists an automorphism taking $a$ to $b$, then $G$ is a $C_p$ vector space, where $C_p$ is the group of prime order $p$.







My question is if an additional hypothesis that $G$ is abelian is needed.




I cannot seem to prove that $G$ is abelian from the hypotheses.



Of course, the point is that all the elements of $G$ have the same prime order $p$.



But I cannot seem to get the result without showing $G$ is abelian, in which case the normality of all subgroups gives what I want.







linear-algebra finite-groups abelian-groups group-homomorphism






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 28 '18 at 23:51









Shaun

8,805113680




8,805113680










asked Oct 28 '14 at 1:39









user187877

162




162












  • If the additional hypothesis is not necessary, then please don't answer the question, by the way.
    – user187877
    Oct 28 '14 at 1:43










  • Are you sure you can't show that the group is already abelian by your hypothesis?
    – Jonny
    Oct 28 '14 at 1:46










  • Can you? I gave it a try but not too hard of a try.
    – user187877
    Oct 28 '14 at 1:48










  • I just saw this result online in the case of abelian groups. But my professor phrased it this way, and I thought he was just trying to be sneaky about saying every element has the same order. But now I'm not so sure.
    – user187877
    Oct 28 '14 at 1:49






  • 2




    These hypotheses can show that $G$ is abelian, so this solves the problem.
    – user187877
    Oct 28 '14 at 2:24


















  • If the additional hypothesis is not necessary, then please don't answer the question, by the way.
    – user187877
    Oct 28 '14 at 1:43










  • Are you sure you can't show that the group is already abelian by your hypothesis?
    – Jonny
    Oct 28 '14 at 1:46










  • Can you? I gave it a try but not too hard of a try.
    – user187877
    Oct 28 '14 at 1:48










  • I just saw this result online in the case of abelian groups. But my professor phrased it this way, and I thought he was just trying to be sneaky about saying every element has the same order. But now I'm not so sure.
    – user187877
    Oct 28 '14 at 1:49






  • 2




    These hypotheses can show that $G$ is abelian, so this solves the problem.
    – user187877
    Oct 28 '14 at 2:24
















If the additional hypothesis is not necessary, then please don't answer the question, by the way.
– user187877
Oct 28 '14 at 1:43




If the additional hypothesis is not necessary, then please don't answer the question, by the way.
– user187877
Oct 28 '14 at 1:43












Are you sure you can't show that the group is already abelian by your hypothesis?
– Jonny
Oct 28 '14 at 1:46




Are you sure you can't show that the group is already abelian by your hypothesis?
– Jonny
Oct 28 '14 at 1:46












Can you? I gave it a try but not too hard of a try.
– user187877
Oct 28 '14 at 1:48




Can you? I gave it a try but not too hard of a try.
– user187877
Oct 28 '14 at 1:48












I just saw this result online in the case of abelian groups. But my professor phrased it this way, and I thought he was just trying to be sneaky about saying every element has the same order. But now I'm not so sure.
– user187877
Oct 28 '14 at 1:49




I just saw this result online in the case of abelian groups. But my professor phrased it this way, and I thought he was just trying to be sneaky about saying every element has the same order. But now I'm not so sure.
– user187877
Oct 28 '14 at 1:49




2




2




These hypotheses can show that $G$ is abelian, so this solves the problem.
– user187877
Oct 28 '14 at 2:24




These hypotheses can show that $G$ is abelian, so this solves the problem.
– user187877
Oct 28 '14 at 2:24










1 Answer
1






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oldest

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1














We may assume $G$ is not trivial. As mentioned in the statement of the problem, it is quick to go from the hypotheses to the fact that $G$ is a $p$-group. Now, every nontrivial $p$-group has a nontrivial center. Let $z$ be some nonzero element of the center. Given $g ,h in G$, by hypothesis we may choose an automorphism $phi$ carrying $g$ to $z$. Then, $phi(gh) = phi(g)phi(h) = phi(h)phi(g) = phi(hg)$. Since $phi$ is injective, we see that $gh = hg$. So, $G$ is abelian.






share|cite|improve this answer





















  • Note that for a group automorphism $phi$, the order of $g$ = order of $phi(g)$. So what can be said about the order of G?
    – Joel Pereira
    Nov 29 '18 at 3:49











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1 Answer
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active

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









1














We may assume $G$ is not trivial. As mentioned in the statement of the problem, it is quick to go from the hypotheses to the fact that $G$ is a $p$-group. Now, every nontrivial $p$-group has a nontrivial center. Let $z$ be some nonzero element of the center. Given $g ,h in G$, by hypothesis we may choose an automorphism $phi$ carrying $g$ to $z$. Then, $phi(gh) = phi(g)phi(h) = phi(h)phi(g) = phi(hg)$. Since $phi$ is injective, we see that $gh = hg$. So, $G$ is abelian.






share|cite|improve this answer





















  • Note that for a group automorphism $phi$, the order of $g$ = order of $phi(g)$. So what can be said about the order of G?
    – Joel Pereira
    Nov 29 '18 at 3:49
















1














We may assume $G$ is not trivial. As mentioned in the statement of the problem, it is quick to go from the hypotheses to the fact that $G$ is a $p$-group. Now, every nontrivial $p$-group has a nontrivial center. Let $z$ be some nonzero element of the center. Given $g ,h in G$, by hypothesis we may choose an automorphism $phi$ carrying $g$ to $z$. Then, $phi(gh) = phi(g)phi(h) = phi(h)phi(g) = phi(hg)$. Since $phi$ is injective, we see that $gh = hg$. So, $G$ is abelian.






share|cite|improve this answer





















  • Note that for a group automorphism $phi$, the order of $g$ = order of $phi(g)$. So what can be said about the order of G?
    – Joel Pereira
    Nov 29 '18 at 3:49














1












1








1






We may assume $G$ is not trivial. As mentioned in the statement of the problem, it is quick to go from the hypotheses to the fact that $G$ is a $p$-group. Now, every nontrivial $p$-group has a nontrivial center. Let $z$ be some nonzero element of the center. Given $g ,h in G$, by hypothesis we may choose an automorphism $phi$ carrying $g$ to $z$. Then, $phi(gh) = phi(g)phi(h) = phi(h)phi(g) = phi(hg)$. Since $phi$ is injective, we see that $gh = hg$. So, $G$ is abelian.






share|cite|improve this answer












We may assume $G$ is not trivial. As mentioned in the statement of the problem, it is quick to go from the hypotheses to the fact that $G$ is a $p$-group. Now, every nontrivial $p$-group has a nontrivial center. Let $z$ be some nonzero element of the center. Given $g ,h in G$, by hypothesis we may choose an automorphism $phi$ carrying $g$ to $z$. Then, $phi(gh) = phi(g)phi(h) = phi(h)phi(g) = phi(hg)$. Since $phi$ is injective, we see that $gh = hg$. So, $G$ is abelian.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Oct 29 '14 at 2:59









user187877

162




162












  • Note that for a group automorphism $phi$, the order of $g$ = order of $phi(g)$. So what can be said about the order of G?
    – Joel Pereira
    Nov 29 '18 at 3:49


















  • Note that for a group automorphism $phi$, the order of $g$ = order of $phi(g)$. So what can be said about the order of G?
    – Joel Pereira
    Nov 29 '18 at 3:49
















Note that for a group automorphism $phi$, the order of $g$ = order of $phi(g)$. So what can be said about the order of G?
– Joel Pereira
Nov 29 '18 at 3:49




Note that for a group automorphism $phi$, the order of $g$ = order of $phi(g)$. So what can be said about the order of G?
– Joel Pereira
Nov 29 '18 at 3:49


















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