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I tried the two ways, but both are all failed. First, counting the order of union of all Sylow $p$, $q$-subgroup. Second, group action from orginal group to set of cosets by Sylow subgroup.

Is there any more tools for proving problems like this?

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2 Answers 2

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First, if $p=q$, then it is the $p$-group of $p^{m+2}$, in which $p$-group of $p^{m+1}$ is the normal subgroup. So assume $p<q$. Similarly assume $m>0$.

We prove that the given group $G$ must have a nontrivial normal subgroup. We consider following two cases.

(1). $m=1$

By Sylow theorem, the number of Sylow $q$-subgroup $n_q|1,\:p,\:p^2$, and $n_q\equiv1\mod q$.

  1. If $n_q=1$, then there is only one Sylow $q$-subgroup which is normal.

  2. If $n_q=p$, then since $p<q,\:n_q\not\equiv1\mod q$. So this is impossible.

  3. If $n_q=p^2$, then there are $p^2$ Sylow $q$-subgroup and at least $2$ Sylow $p$-subgroup (for otherwise Sylow $p$-subgroup would be normal). Let $H,K$ be any 2 distinct Sylow $q$-subgroups and $|H|=|K|=q$. Since $H\cap K<H$, $|H\cap K|=1$ or $q$. If $|H\cap K|=q$, then $H=K$, which is impossible. So $|H\cap K|=1$. Similarly let $H,K$ be 2 distinct Sylow $p$-subgroups and $|H|=|K|=p^2$. Then $|H\cap K|=1,\:p$ or $p^2$. If $|H\cap K|=p^2$, then $H=K$, which is impossible. So $|H\cap K|\leqslant p$. So there are at least $$p^2(q-1)+2(p^2-1)-(p-1)=p^2q+p^2-p-1$$ elements in the group which is more than the given group with order of $p^2q$, which is impossible.

(2). $m>1$

Let $H,K$ be 2 distinct Sylow $q$-subgroups. Suppose $|H\cap K|<q^{m-1}$. Then $$ |HK|=\frac{|H||K|}{|H\cap K|}>q^{m+1} $$ But since $|HK|\leqslant |G|=p^2q^{m}$, this is impossible. So $|H\cap K|=q^{m-1}$.

By Normal subgroup of prime index, since $q$ is the smallest prime index dividing $q^m$ $$ H\cap K\vartriangleleft H \quad \text{and}\quad H\cap K\vartriangleleft K $$ Thus $$ H\subset N_G(H\cap K) \quad \text{and}\quad K\subset N_G(H\cap K) $$ where $N_G(H\cap K)$ is the normalizer of $H\cap K$ in $G$. So $$ |N_G(H\cap K)|\geqslant |HK|=\frac{|H||K|}{|H\cap K|}\geqslant q^{m+1} $$ Note that $HK$ may not be a subgroup but $N_G(H\cap K)$ is.

$|N_G(H\cap K)|=q^{m+1}$ is impossible since $N_G(H\cap K)<G$ and $|N_G(H\cap K)|\nmid|G|$.

If $|N_G(H\cap K)|>q^{m+1}$, then $|N_G(H\cap K)|\geqslant pq^{m+1}$. But $$ |N_G(H\cap K)|>p^2q^{m}=|G| $$ which is impossible. So $H\cap K\vartriangleleft G$.

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    $\begingroup$ How do you justify two $q$-Sylow subgroups intersect trivially? $\endgroup$ Aug 9, 2015 at 2:36
  • $\begingroup$ What if $q=2,p=3$? Then $n_2 = 3$ has $3 = 1 \mod 2$. $\endgroup$
    – nullUser
    Aug 9, 2015 at 2:38
  • $\begingroup$ $p<q$ is assumed. $\endgroup$ Aug 9, 2015 at 2:43
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The cases where $m=0$ or $q=p$ are already discussed in the other post. So suppose $q>p$ and $m\geq 1$.

If $m=1$: Then any two $q$-Sylow subgroups must intersect trivially, so if $n_q = p^2$, then there must be $p^2(q-1) + 1$ elements in the group. Also, $n_q \neq p$ since $p\neq 1\pmod{q}$. So $n_q = 1$ the $q$-Sylow subgroup is normal.

If $m>1$: Suppose $P_1$ and $P_2$ are two distinct $q$-Sylow subgroups, then set $H=P_1\cap P_2$, then $$ p^2q^m \geq |P_1P_2| = |P_1||P_2|/|H| = q^{2m}/|H| \Rightarrow |H| \geq \frac{q^m}{p^2} > q^{m-2} $$ Since $|H| \mid q^m$ and $|H| < q^m$, it follows that $|H| = q^{m-1}$, and so $H$ is normal in both $P_1$ and $P_2$. Furthermore, $$ |P_1P_2| = q^{2m}/q^{m-1} = q^{m+1} $$ Now, $P_1$ and $P_2$ are both contained in $N_G(H)$, the normalizer of $H$ in $G$. Hence, $$ P_1P_2 \subset N_G(H) \Rightarrow |N_G(H)| \geq q^{m+1} $$ But $|N_G(H)| \mid p^2q^m$, so $|N_G(H)| = p^2q^m$, and so $H \vartriangleleft G$.

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