SYLOW PAIRS IN FINITE GROUPS

Consider a finite group and let denote a prime number. A Sylow subgroup of is a subgroup of whose order is as large as is allowed by Lagrange’s theorem, and is the set all such subgroups. The essential theorem of group theory asserts that Sylow subgroups always exist and mod . In this note, we say that an ordered pair is a Sylow pair if there exists group with , where is an integer. We prove that the ordered pair (7,15) is not a Sylow pair.