Equivalence relation

Let $ G $ be a finite group and $ \pi $ a subset of the prime numbers that divide the order $ n $ of $ G $. A Sylow $ \pi $- basis $ S $ is a collection of Sylow $ p $- subgroups $ P _ {p} $ of $ G $( cf. Sylow subgroup), one for each prime $ p $ in $ \pi $, such that: If $ P _ {p _ {1} } \dots P _ {p _ {r} } $ are in $ S $, then the order of every element in $ \{ G _ {p _ {1} } \dots G _ {p _ {r} } \} $( the subgroup generated by $ P _ {p _ {1} } \dots P _ {p _ {r} } $) is a product of non-negative powers of $ p _ {1} \dots p _ {r} $. If $ \pi $ is the set of all primes dividing $ n $, one speaks of a complete Sylow basis. Two Sylow bases are conjugate if there is a single element of $ G $ that by conjugation transforms all the groups of the first into those of the second. Hall's second theorem, [a2], says that every finite solvable group has a complete Sylow basis, and that all these bases are conjugate. Conversely, if a finite group has a complete Sylow basis, then it is solvable (cf. also Solvable group).

References

[a1]	A.G. Kurosh, "The theory of groups" , 2 , Chelsea (1960) pp. 195ff (Translated from Russian)
[a2]	P. Hall, "On the Sylow systems of a soluble group" Proc. London Math. Soc. , 43 (1937) pp. 316–323