When do Sylow subgroups of the same order have trivial intersection?
I'm curious because I recently read a proof where it was computed that there are $8$ Sylow 7-subgroups, and hence $8\cdot 6=48$ elements of order 7. This seems to assume that each of the Sylow subgroups has trivial intersection.
Why is this the case? Is it true even when the Sylow groups have order a prime power, not necessarily just prime order?
Best Answer
Two subgroups of order $7$ are either equal or intersect trivially. Indeed, their intersection is a subgroup of each of them, and they have exactly two subgroups.
In general, a group $G$ can very well have many $p$-Sylow subgroups intersecting non-trivially. For example, let $H$ be any group which does not have a normal $p$-Sylow, and consider the group $G=\mathbb Z_p\times H$.