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$|G|=105=3\cdot 5\cdot 7$. We know that the Sylow $5-$subgroup $P_5$ must be normal and the Sylow $7-$subgroup $P_7$ must also be normal by simply counting elements. With the additional assumption of the Sylow $3-$subgroup $P_3$ being normal then we can say:

1.) $|G/C_G(P_3)|$ must divide both $|\operatorname{Aut}(P_3)|=2$ and $|G|=105$. This means that $$|G/C_G(P_3)|=1\quad\Longrightarrow\quad G=C_G(P_3)\quad\Longrightarrow\quad P_3\le Z(G).$$

2.) $|G/C_G(P_5)|$ must divide both $|\operatorname{Aut}(P_5)|=4$ and $|G|=105$. This means that $$|G/C_G(P_5)|=1\quad\Longrightarrow\quad G=C_G(P_5)\quad\Longrightarrow\quad P_5\le Z(G).$$

3.) $|G/C_G(P_7)|$ must divide both $|\operatorname{Aut}(P_7)|=6$ and $|G|=105$. This means that $$|G/C_G(P_7)|=1\quad\Longrightarrow\quad G=C_G(P_7)\quad\Longrightarrow\quad P_7\le Z(G).$$

Thus $\langle P_3,P_5,P_7\rangle\le Z(G)\le G$ and since $|\langle P_3,P_5,P_7\rangle|=105$, we must $|Z(G)|=105$ and we conclude $Z(G)=G$ and that $G$ is abelian.

Do I have the just of this right or am I making a conceptual error somewhere?

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  • $\begingroup$ If all of the $p$-Sylow subgroups of $G$ are normal, then $G$ is isomorphic to the direct product of its $p$-Sylow subgroups. $\endgroup$ Feb 22, 2016 at 4:18
  • $\begingroup$ Right. But the above argument proves the specific case, correct? I am reading Dummit and Foote and all the direct product theorems are in the chapter after the Sylow theorems. Just making sure I can prove it with what I am supposed to know up to this point. $\endgroup$ Feb 22, 2016 at 4:30
  • $\begingroup$ @Ethan Alwaise: If the Sylow 3-subgroup is always normal, then every group of order 105 would be abelian. See this for a demonstration of a nonabelian group of 105: math.stackexchange.com/questions/151767/… $\endgroup$ Feb 22, 2016 at 4:40
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    $\begingroup$ Well this is an old question, I believe. But I think there is something confusing in your 3.) derivation: $|G/C_{G}(P_7)|$ could be $3$, dividing both $6$ and $105$. Then you may not have that $G = C_G(P_7)$. $\endgroup$
    – mathdoge
    Jul 17, 2019 at 4:21
  • $\begingroup$ @mathdoge - I haven't looked at this stuff in three years, but I think the solution is that we already know the Sylow 7-subgroup is normal, so that makes the possibility of $|G/C_G(P_7)|=3$ impossible. $\endgroup$ Jul 19, 2019 at 14:02

2 Answers 2

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If $G/Z(G)$ is cyclic, then $G$ is abelian and if $C_{G}(H) = G$, then $H \leq Z(G)$. Since $H$ is normal, let $G$ act on $H$ via conjugation. This action induces a homomorphism $\phi: G \rightarrow \text{Aut}(H) = \mathbb{Z}_{2}$ with $\ker \phi = C_{G}(H)$.

Since the order of $G$ is odd, we must have $|G/C_{G}(H)| = 1 \implies G = C_{G}(H) \implies H \leq Z(G)$. It follows that $|G/Z(G)| \in \{35, 5, 7\}$ and it follows that $G/Z(G)$ is cyclic.

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This all looks correct to me. It's worth checking out the hint in the comments for a quicker proof, but this technique is good.

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    $\begingroup$ I agree. See my response for why I didn't do that. $\endgroup$ Feb 22, 2016 at 4:30
  • $\begingroup$ It's good practice. Out of curiosity, are you a college student? If so, what year are you $\endgroup$ Feb 22, 2016 at 4:31
  • $\begingroup$ I am a PhD student. I am trying to prepare for my qualifying exam in Algebra right now. I finished the course sequence last spring, but had to miss out on taking it last summer due to my health. I am back to healthy now and am trying to pick up the pieces from last year, so to speak! $\endgroup$ Feb 22, 2016 at 4:34
  • $\begingroup$ @LaarsHelenius Gotcha. I recently became aware my university teaches things unusually, so I've become curious about the standard pacing. $\endgroup$ Feb 22, 2016 at 4:35
  • $\begingroup$ In our year long course sequence preparing for the qualifier, we studied the first 14 chapters of Dummit and Foote. Fairly standard stuff for a year long graduate course in Algebra I think. $\endgroup$ Feb 22, 2016 at 4:37

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