Question: are sylow p subgroups disjoint?
Answer: By Lagrange, distinct subgroups of size p must intersect trivially, so their nonidentity elements are disjoint from each other. Therefore each subgroup of size p has its own p − 1 elements of order p, not shared by the other subgroups of size p. The number of elements of order p is therefore k(p − 1). Theorem 5.4.