Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$

Solution 1:

Let $R = P\cap Q$, where $\;P\leq G$ a p-sylow subgroup and $Q\leq G$ is a q-sylow subgroup, and $p\neq q$.

Then for any $r \in R$, $r$ has order equal a power of $p$ and equal to a power of $q$.

What must $R$ "look like"?

The same can not necessarily be said about the case $p = q$, for $\;p, q$ prime.

Solution 2:

The intersection of two different Sylow $p$-subgroups for the same prime $p$ is less predictable. If a finite group $G$ has a normal $p$-subgroup $U,$ then $U$ is contained in any Sylow $p$-subgroup of $G.$ It is quite difficult to give precise conditions which will guarantee the existence of two Sylow $p$-subgroups $P_{1}$ and $P_{2}$ such that $P_{1} \cap P_{2} = U.$ A theorem of J. Brodkey asserts that there are such subgroups if $G/U$ has Abelian Sylow $p$-subgroups. ut, for example there is a finite group $G$ with $|G| = 144$ such that $G$ has no non-identity normal $2$-subgroup, but there do not exist Sylow $2$-subgroups $P_{1}$ and $P_{2}$ of $G$ with $P_{1} \cap P_{2} = 1.$ ( because $|G| < |P_{1}||P_{2}|$). The group $G$ in this example has a semi-dihedral Sylow $2$-subgroup.

Solution 3:

We know that the intersection of two Sylow $p$-subgroups for the same prime $p$ is a prime power. Without more data, there is really nothing more to say. In fact, for any integer $n \geq 1$ we can find a finite group $G$ with the following properties:

1. The Sylow $p$-subgroups of $G$ have order $p^n$.
2. For any $0 \leq i \leq n$, there exists Sylow $p$-subgroups $P$ and $Q$ such that $P \cap Q$ has order $p^i$.

To construct such $G$, first let $q$ be a prime such that $q \equiv 1 \mod{p}$ and $q > p$ and $q > n$. The existence of such $q$ follows from Dirichlet's theorem.

Let $T$ be the nonabelian group of order $pq$. Define $G = T \times T \times \ldots \times T$, where $T$ appears in the direct product exactly $n$ times. Now $T$ has exactly $q$ Sylow $p$-subgroups, so let $P_1, P_2, \ldots, P_n, P$ be distinct Sylow $p$-subgroups of $T$. Let

\begin{align*} Q_0 &= P \times P \times P \times \ldots \times P \\ Q_1 &= P_1 \times P \times P \times \ldots \times P \\ Q_2 &= P_1 \times P_2 \times P \times \ldots \times P \\ \ldots \\ Q_{n-1} &= P_1 \times P_2 \times P_3 \times \ldots \times P_{n-1} \times P \\ Q_n &= P_1 \times P_2 \times P_3 \times \ldots \times P_{n-1} \times P_n \end{align*}

Now each $Q_i$ is a Sylow $p$-subgroup of $G$ and $|Q_i \cap Q_n| = p^i$ for all $0 \leq i \leq n$.