Sylow basis

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