Pi-separable group

A group which has a normal series such that the order of every factor contains at most one prime from ( is a set of prime numbers). The class of -separable groups contains the class of -solvable groups (cf. -solvable group). For finite -separable groups, the -Sylow properties (cf. Sylow theorems) have been shown to hold (see [1]). In fact, for any set , a finite -separable group contains a -Hall subgroup (cf. also Hall subgroup), and any two -Hall subgroups are conjugate in . Any -subgroup of a -separable group is contained in some -Hall subgroup of (see [2]).

References

[1]	S.A. Chunikhin, "On -separable groups" Dokl. Akad. Nauk SSSR , 59 : 3 (1948) pp. 443–445 (In Russian)
[2]	P. Hall, "Theorems like Sylow's" Proc. London Math. Soc. , 6 : 22 (1956) pp. 286–304

Comments

Chunikhin's theorem says that if is a divisor of the order of a -separable group such that , , and if all prime divisors of are in , then has a subgroup of order and all these subgroups are conjugate in . If consists of all prime numbers this becomes Hall's first theorem.

Gol'berg's theorem, [a2], says that if is a finite -separable group and is a subset of , then has a Sylow -basis (cf. Sylow basis) and all these bases are conjugate.

References

[a1]	A.G. Kurosh, "The theory of groups" , 2 , Chelsea (1956) pp. 195ff (Translated from Russian)
[a2]	P.A. Gol'berg, "Sylow bases of -separable groups" Dokl. Akad. Nauk SSSR , 60 (1949) pp. 615–618 (In Russian)
[a3]	S.A. Chunikhin, "On -properties of finite groups" Mat. Sb. , 25 (1949) pp. 321–346 (In Russian)