Sylow basis

Let be a finite group and a subset of the prime numbers that divide the order of . A Sylow -basis is a collection of Sylow -subgroups of (cf. Sylow subgroup), one for each prime in , such that: If are in , then the order of every element in (the subgroup generated by ) is a product of non-negative powers of . If is the set of all primes dividing , one speaks of a complete Sylow basis. Two Sylow bases are conjugate if there is a single element of 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