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I can't seem to make progress on proving the following: if $G$ finite, $P < G$ a subgroup of order $p$ and $P = C_G(P)$ then $P$ is a Sylow $p$-subgroup of $G$. $C_G(P)$ is the centralizer of $P$ in $G$, i.e. the set $\{a \in G : ax=xa \ \forall x \in P\}$.

My work so far: Since $P = C_G(P)$ then $P$ abelian. I think to show $P$ is a Sylow p-subgroup we need to show that if $|G| =mp^n$ for some $m$ coprime to $p$ and $n > 0$, then $n = 1$ (so $P$ has maximal order). There should also exist $Q$ a Sylow p-subgroup of $G$ such that $P < Q$. If we can show $P = Q$ then we are done.

I suspected a counting argument using the class equation: $|G| = |Z(G)| + \sum[G:C_G(x_i)]$ where $x_i \in G \setminus Z(G)$. If $x_i \in P$ then $P = C_G(P) \subset C_G(x_i)$ so $p$ divides $|C_G(x_i)| = p^{k_i}$ for $1 \le k_i \le n$. Then \begin{align} |G| &= |Z(G)| + \sum[G:C_G(x_i)] \\ mp^n &=|Z(G)| + |G\setminus Z(G)|p^{k_i}. \end{align} I'm not sure if I'm heading in the right direction since I don't see where we would even get a contradiction here. Thanks for any feedback.

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As you say, let $Q$ be a $p$-Sylow with $P\le Q$. Applying the class equation to $Q$ shows that $Q$ has a non-trivial center, since all non-central elements have class sizes divisible by $p$, so $|Z(Q)|=1$ would imply $p$ divides $p^n-1$, which is absurd. Since $P=C_G(P)$ and since $Z(Q)$ centralizes $P$, we must have $Z(Q)\le P$ and thus $Z(Q)=P$ since $P$ has order $p$ and $Z(Q)$ has order at least $p$. But then $Q\le C_G(Z(Q))=C_G(P)=P$, so $Q\le P$. Thus $P$ is a $p$-Sylow.

If you know the full statement of Sylow's theorem, there's an even shorter proof: Since the full theorem states that any $p$-subgroup that is not Sylow is an index $p$-subgroup of some other subgroup, then if $P$ is not Sylow, $P<R$ with $|R|=p^2$. But every group of order $p^2$ is abelian and thus $R$ centralizes $P$. Contradiction.

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  • $\begingroup$ For showing $P = Z(Q) $ is it correct to simply state that $P < Z(Q) < C_G(P)$? I was a bit confused on the phrase "$Z(Q)$ centralizes $P$". Thank you for the response. $\endgroup$ – Junkhook9000 Jun 22 '18 at 0:31
  • $\begingroup$ No, you would want to say $Z(Q)=C_Q(Q)\le C_G(Q)\le C_G(P)=P$ $\endgroup$ – C Monsour Jun 22 '18 at 1:18
  • $\begingroup$ I see, thank you for explaining. $\endgroup$ – Junkhook9000 Jun 22 '18 at 15:57
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I like the short proof of C Monsour, +1 from me. Here is another proof. Actually two of them. Observe that $N_G(P)/C_G(P)$ can be homomorphically embedded in $Aut(P)$. Since $P$ is cyclic of prime order $p$, $Aut(P) \cong C_{p-1}$, hence $p$ does not divide $|N_G(P):P|$.

There are now at least two ways to finish the proof.

(i) Since $P$ is a $p$-group it must be contained in some $Q \in Syl_p(G)$, and assume $P \subsetneq Q$. Since normalizers grow (see for example I.M. Isaacs Finite Group Theory Theorem 1.22), we must have $P \subsetneq N_Q(P)$, whence there is a $x \in Q-P$ normalizing $P$. Now look at $\bar{x}$ as element of $N_G(P)/P$. $\bar{x}$ is a $p$-element (since $x \in Q$) and $p$ does not divide $|N_G(P)/P|$. Hence $\bar{x}=\bar{1}$, contradicting $x \notin P$. So $P=Q$.

(ii) In general, if $P$ is a $p$-subgroup of a group $G$, then $|G:P| \equiv |N_G(P):P|$ mod $p$ (for a proof: let $G$ act by left multiplication on the left cosets of $P$ in $G$, and apply the Orbit-Stabilizer counting theorem (or see here)). We already saw that $p \nmid |N_G(P):P|$, hence it follows that $p \nmid |G:P|$, meaning $P$ is Sylow.

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    $\begingroup$ Excellent - as always! $\endgroup$ – Moritz Jun 22 '18 at 14:37
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    $\begingroup$ This is really enlightening thank you. I am still working on understanding all the points, but my first question is: can you elaborate on the homomorphic embedding of $N_G(P)/C_G(P) $ in $Aut(P) $? $\endgroup$ – Junkhook9000 Jun 22 '18 at 16:08
  • $\begingroup$ In general, if $H \leq G$, then define a map $\alpha$ from $N_G(H)$ to $Aut(H)$ as follows: $\alpha(x)=\phi_x$, with $x \in N_G(H)$, and where $\phi_x$ is defined as $\phi_x(h)=xhx^{-1}$ for all $h \in H$. Show three things: $\alpha$ is well-defined, $\alpha$ is a homomorphism, and finally, $ker(\alpha)=C_G(H)$. Now apply the First Isomorphism Theorem. This gives you an embedding $N_G(H)/C_G(H) \hookrightarrow Aut(H)$. It is sometimes called the "N/C" Theorem. $\endgroup$ – Nicky Hekster Jun 22 '18 at 17:31
  • $\begingroup$ @Moritz thank you so much for the compliment! $\endgroup$ – Nicky Hekster Jun 22 '18 at 17:37
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    $\begingroup$ Thanks for elaborating. This really helped my understanding. $\endgroup$ – Junkhook9000 Jun 23 '18 at 2:36

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