Let $G$ be a finite group and assume it has more than one Sylow $p$-subgroup.
It is known that order of intersection of two Sylow p-subgroups may change depending on the pairs of Sylow p-subgroups.
I wonder whether there is a condition which guarantees that intersection of any two Sylow $p$-subgroups has the same order.
Thanks for your help.
Best Answer
I can prove the following, which provides a criterion to start with.
Theorem Let $G$ be a finite group. Then the following are equivalent.
$(a)$ For all $S,T \in Syl_p(G)$ with $S \neq T$, $S \cap T=1$.
$(b)$ For each non-trivial $p$-subgroup $P$ of $G$, $N_G(P)$ has a unique Sylow $p$-subgroup.
$(c)$ For each non-trivial $p$-subgroup $P$ of $G$, $P$ is contained in a unique Sylow $p$-subgroup of $G$.