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 |
Sylow basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sylow_basis&oldid=15735