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I am trying to prove that a group of order $p^2q^2$ where $p$ and $q$ are primes is solvable, without using Burnside's theorem. Here's what I have for the moment:

  • If $p = q$, then $G$ is a $p$-group and therefore it is solvable.

  • If $p \neq q$, we shall look at the Sylow $p$-subgroups of $G$. We know from Sylow's theorems that $n_p \equiv 1 \pmod p$ and $n_p \mid q^2$, therefore $n_p \in \{1, q, q^2\}$.

  • If $n_p = 1$, it is over, because the Sylow $p$-Subgroup $P$ is normal in $G$ of order $p^2$, and $G/P$ has order $q^2$. Thus both are solvable and $G$ is solvable.

  • If $n_p = q^2$, we have $q^2(p^2-1)$ elements of order $p$ or $p^2$ in $G$, and we have $q^2$ elements left to form a unique Sylow $q$-subgroup. By the same argument as before, $G$ is solvable.
  • That's where I'm in trouble. I don't know what to do with $n_p = q$. It seems to lead nowhere.

Thanks in advance for any help!

Laurent

Laurent Hayez
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    Welcome to math.SE! Thanks for showing your work so far! – Qiaochu Yuan Jun 15 '14 at 16:19
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    Since you reduced this to $n_p=q$, you have $n_q = p$ as well. Also $p<q$ without loss of generality. – Myself Jun 15 '14 at 16:23
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    I'm not sure that I follow your argument. You seem to be assuming that distinct Sylow $p$-subgroups intersect in the identity, and that needs some justification at least, as far as I can see. This can't hold for both a Sylow $p$-subgroup and a Sylow $q$-subgroup in any case. If $P \in {\rm Syl}{p}(G)$ has $|P| >|Q|$ for $Q \in {\rm Syl}{p}(G),$ for example, we have $P \cap P^{g} >1$ for any $g \in G \backslash N_{G}(P),$ otherwise $|PP^{g}| > |G|.$ – Geoff Robinson Jun 15 '14 at 16:50
  • @GeoffRobinson What I implicitly used is that the order of an element divides the order of the group. That means, the elements in the $p$-Sylows have order $1$, $p$ or $p^2$ and none of them divide $q^2$ (unless $p = q$ but I supposed that's not the case here). Thus if I have $q^2$ $p$-Sylows, I used $q^2(p^2-1)$ (I do not count the identity element) elements that do not divide $q^2$. As there exists at least one $q-Sylow$ of order $q^2$, I must use the remaining elements to form this $q$-Sylow, which is normal as it is unique. And yes, that's why I assume $P \cap Q = {1}$. – Laurent Hayez Jun 15 '14 at 18:25
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    Yes, but what worries me is that you could have elements of order $p$ or $p^{2}$ which lie in more than one Sylow $p$-subgroup, so that $q^{2}(p^{2}-1)$ would be an overcount of the number of elements of order $p$ or $p^{2}.$ – Geoff Robinson Jun 15 '14 at 18:41
  • I see your point. I thought that Sylow $p$-subgroups only had identity element in common. In fact I thought that because the second Sylow's theorem says that two Sylow $p$-subgroups are conjugate. Then how can I be sure that the Sylow $p$-subgroups only have the idententy in common? Do you have such an example? Thanks for noticing it. – Laurent Hayez Jun 15 '14 at 19:00
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    In a group whose Sylow $p$-subgroup has prime order $p,$ it is clear that distinct Sylow $p$-subgroups only have the identity in common. In general, for larger Sylow subgroups that need not be the case. In particular groups, it needs to be checked. – Geoff Robinson Jun 15 '14 at 22:32

3 Answers3

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You argument works just as well with $p$ and $q$ switched, so the only time you have trouble is if both $n_p=q$ and $n_q = p$. Since $1\equiv n_p \mod p$ and $1\equiv n_q \mod q$ this puts very strong requirements on $p$ and $q$.

Hint 1:

Unless $n_p=1$, $n_p > p$.

Hint 2:

If $n_p=q$, then $q>p$. If $n_q =p$, then $p>q$. Oops.

Fix for OP's argument:

The OP's argument is currently flawed in the case $n_p=q^2$, so this answer is only truly helpful after that flaw is fixed.

A very similar argument to the one given in this answer works. First part of your argument works, and the $p-q$ symmetry helps:

If $n_p=1$ or $n_q=1$, then the group is solvable.

Now we use the Sylow counting again to get some severe restrictions:

If $n_p \neq 1$, then $n_p \in \{q,q^2\}$ and in both cases we have $1 \equiv q^2 \mod p$. Similarly, if $n_q \neq 1$, then $1 \equiv p^2 \mod q$.

Unfortunately now we don't get an easy contradiction, but at least we only get one possibility:

Since $p$ divides $q^2-1 = (q-1)(q+1)$, we must also have $p$ divides $q-1$ or $q+1$, so $p \leq q+1$ and $q \leq p+1$, so $p-1 \leq q \leq p+1$. If $p=2$ is even, then $q$ is trapped between 1 and 3, so $q=3$. If $p$ is odd, then $p-1$ and $p+1$ are both even, so the only possibility for $q \neq p$ is $q=p-1=2$ (so $p=3$) or $q=p+1=2$ (so $p=1$, nope). Hence the only possibility is $p=2$ and $q=3$ (or vice versa).

In this case, we get:

If $p=2$ and $q=3$, then $n_q \in \{2,4\}$. Considering the permutation action of $G$ on its Sylow $q$-subgroups, we know that $n_q=2$ is impossible (Sylow normalizers are never normal) and $n_q=4$ means $G$ has a normal subgroup $K$ so that $G/K$ is isomorphic to a transitive subgroup of $S_4$ containing a non-normal Sylow 3-subgroup and having order a divisor of 36. The only such subgroup is $A_4$, so $K$ has order 3. Hence $G/K\cong A_4$ and $K \cong A_3$ are solvable, so $G$ is solvable.

Jack Schmidt
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    I am tempted to henceforth end all my proofs by contradiction with "oops". +1 – RghtHndSd Jun 15 '14 at 18:19
  • Thank you! I think I managed to solve it with your hints! As you said, if we have $n_p = q$ and $n_q = p$, we fall on the contradiction you mentioned in the hint 2, $p > q$ and $q > p$. Thus we can conclude that this case can't happen, and therefore $G$ is solvable. Is that right? – Laurent Hayez Jun 15 '14 at 18:20
  • That's OK, but unfortunately your reduction to the case $n_p=q$ and $n_q=p$ is not fully justified. – Derek Holt Jun 15 '14 at 19:07
  • I gave a fix for the OP's reduction. It is probably harder than it should be. :-) – Jack Schmidt Jun 15 '14 at 19:48
  • I get almost everything, but the last part of the argument goes a bit beyond my current level, I think. I need a few precision: 1. as english is not my mother tongue, I don't know what an OP's reduction is. 2. $n_q= 2$ is impossible because Sylow $q$-subgroups are conjugate, and not normal? 3. I do not get the last part of the argument (since $n_q = 4$). I think I don't know enough theory for the moment, isn't there any simpler way? – Laurent Hayez Jun 15 '14 at 20:04
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    @LaurentHayez: "OP's reduction" means "The argument Laurent Hayez gave in the original question that reduced to the case $n_p=q^2$". I'm sorry about the very last hint (it basically says "Which subgroups of $S_4$ and $S_2$ actually describe the action of a group on its Sylow 3-subgroups. The only two permutation groups that work are $S_4$ and $A_4$, and only $A_4$ has the right order.) Both Geoff Robinson's and Mikko Kornhonen's answers should work, but I think they are probably also a little too hard. Maybe if you combine the easy parts of all of our answers, it will work. – Jack Schmidt Jun 16 '14 at 15:12
  • While trying to prove the first point in the proof given by Mikko Korhonen, I had to prove a very similar thing to what you defined. In fact, the group action induces a homomorphism from $G$ to $S_4$, whose kernel is a normal group in $G$, let's call it $K$. We know $|K|$ divides $|G| = 36$. Now $G/K$ contains a $3$-Sylow because by the isomorphism theorem, $G/K$ is isomorphic to a subgroup of $S_4$ which permutes the $3$-Sylows. If I'm correct until now, I have 2 questions: 1. why is it $A_4$ and not Klein's vierergruppe? 2. Why is $\ker \phi \neq {e}$? – Laurent Hayez Jun 16 '14 at 20:23
  • @LaurentHayez: (1) If $Q$ is a Sylow 3-subgroup of $G$, then $QK/K$ is a Sylow 3-subgroup of $G/K$ and the action of $G/K \leq S_4$ is exactly conjugation on the four conjugates of $QK/K$. In $G/K \cong V_4$, $QK/K = K/K$ does not have four conjugates. (2) $G/K$ has order dividing $|G|=36$ and $|S_4|=24$, so order dividing their gcd, 12. Hence $K$ has order at least 3 (exactly 3 based on (1)). – Jack Schmidt Jun 16 '14 at 20:36
  • Is it true that because $|V_4|=4$, it cannot even contain a Sylow 3-subgroup? If yes that would mean the only remaining subgroup to satisfy $|G/K| \big| |G|$, $|G/K| \big| |S_4|$ and must contain a Sylow 3-subgroup is indeed $A_4$. Thanks again for your explanations and your quick answers! One last question, why is it sufficient to restrain to a group of order $36$? Because clearly, not every groups of order $p^2q^2$ have order $36$. – Laurent Hayez Jun 16 '14 at 21:07
  • Since $|V_4|=4$ its Sylow 3-subgroup is normal (of order 1), so that's no good. 36: Part 3 of my "fix" section shows ${p,q}={2,3}$ so $p^2 q^2=36$. – Jack Schmidt Jun 16 '14 at 21:16
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Assuming that you know that groups of order $p^2q$, $pq$ and $p^k$ are solvable, it is enough to prove that a group of order $p^2q^2$ is not simple.

Suppose that $G$ is a simple group of order $p^2q^2$. By symmetry (and since $p$-groups are solvable) we may assume $p > q$. Steps to reach a contradiction:

  1. Prove the following: if $G$ is a finite group with a subgroup $H$ of index $r$, where $r$ is the smallest prime divisor of $|G|$, then $H$ is a normal subgroup.

  2. By 1. $n_p = q$ cannot happen since $G$ is simple. Therefore $n_p = q^2$.

  3. If there exist distinct Sylow $p$-subgroups $P_1$ and $P_2$ such that their intersection $D = P_1 \cap P_2$ is nontrivial, then $D$ has order $p$. Now $D$ is normal in both $P_1$ and $P_2$, but not normal in all of $G$, so $N_G(D)$ has order $p^2q$. This is a contradiction by 1.

  4. Therefore distinct Sylow $p$-subgroups of $G$ have pairwise trivial intersection. By 2. this means that there are $q^2(p^2-1)$ elements of order $p$ or $p^2$. But then $G$ has a normal Sylow $q$-subgroup.

Mikko Korhonen
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  • Hi! First of all, thanks for helping! I like this solution, nevertheless, if the 2., 3. and 4. points are easy to understand, the first one is quiet hard to prove. I tried to follow the steps of this post: http://math.stackexchange.com/questions/164244/normal-subgroup-of-prime-index?rq=1 I just don't get the argument "since $p$ is the smallest prime that divides $|G|$, it follows that $|G/K|=p$. That would be great if you could explain, because it the only thing that I don't understand. – Laurent Hayez Jun 16 '14 at 20:00
  • @LaurentHayez: Since $p$ is the smallest prime divisor, $\gcd(|G|, p!) = p$. This follows since a common divisor of $|G|$ and $p!$ would have all of its prime divisors $\leq p$, hence it has to be $p$ or $1$. If it is still not clear, you could try proving it in the case $|G| = p^2q^2$. – Mikko Korhonen Jun 16 '14 at 21:00
  • @MikkoKorhonen To clarify why $|N_G(D)|=p^2q$ in step 3 : is this because each of the $2p^2-p$ elements of $P_1 \cup P_2$ belong to the normalizer, and the only factor of $p^2q^2$ greater than $2p^2-p$ is $p^2q$ ? – RDL May 28 '24 at 18:10
  • @RDL: For example. Or note that $|N_G(D)|$ must be a multiple of $p^2$ since it contains $P_1$. So $|N_G(D)| = p^2$, $p^2q$, or $p^2q^2$. Then $|N_G(D)| = p^2$ is not possible since $N_G(D)$ contains both $P_1$ and $P_2$, and $|N_G(D)| = p^2q^2$ is not possible since $G$ is simple. – Mikko Korhonen May 29 '24 at 02:40
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I will give a different series of hints/steps to show the following: if $p >q,$ then $G$ has a non-identity normal $p$-subgroup. Let $P \in {\rm Syl}_{p}(G),$ and suppose that no non-identity subgroup of $P$ is normal in $G$ (that includes $P$ itself, of course). Notice that $q \not \equiv 1$ (mod $p$), since $1 \neq q < p.$ Hence $G$ must have $q^{2}$ Sylow $p$-subgroups, and we must have $q^{2} \equiv 1$ (mod $p$). Hence $p | q+1$ ( we can't have $p|q-1$ as $q <p$). But $q <p,$ so $q+1 \leq p,$ so we must have $q = p-1$. Now $p \neq 2$ as $p>q,$ so $q$ is even. Hence $q = 2$ and $p=3,$ as $p$ is a prime. Hence $|G| = 36$. Now $P = N_{G}(P)$ by Sylow's Theorem. Furthermore, there is non proper subgroup $M$ of $G$ which strictly contains $P.$ For otherwise we would have $[M:P] \equiv 1$ (mod $3$) and $[G:M] \equiv 1$ (mod 3), forcing $[G:P] \geq 16,$ which is not the case. Now let $g^{-1}Pg$ be another Sylow $3$-subgroup of $G.$ Then $P \cap g^{-1}Pg \neq 1$ as $|P||g^{-1}Pg| > |G|$.However $P$ and $g^{-1}Pg$ are both Abelian, so $P \cap g^{-1}Pg \lhd \langle P,g^{-1}Pg \rangle >P.$ But there is no subgroup of $G$ strictly between $P$ and $G,$ so $P \cap g^{-1}Pg \lhd G.$ (actually your (Laurent's) argument works in $G/(P \cap g^{-1}Pg)$ to show that a Sylow $2$-subgroup is normal in that quotient group.

Geoff Robinson
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  • Thank you, I got everything except one point. It is the part "For otherwise we would have $[M:P] \equiv 1 \pmod 3$ and $[G:M] \equiv 1 \pmod 3$". I really can't see why the indexes should be equivalent to $1$ modulo $3$? – Laurent Hayez Jun 16 '14 at 20:04
  • Because $P$ is also the normalizer of a Sylow $3$-subgroup of $M$, we must have $[M:P] \equiv 1$ (mod $3$). Because $P$ is the normalizer of a Sylow $3$-subgroup of $G,$ we must have $[G:P ]\equiv 1$ (mod $3$). Hence we have $[G:P] = [G:M][M:P] \equiv 1$ (mod $3$) also forces $[G:M] \equiv 1$ (mod $3$). – Geoff Robinson Jun 16 '14 at 22:21