Hall subgroup

A subgroup of a finite group whose order is coprime to its index. It is named after Ph. Hall, who in the 1920's initiated the study of such subgroups in finite solvable groups (cf. Solvable group).

In a finite $\pi$-divisible group there is a Hall $\pi$-subgroup (a Hall subgroup whose order is divisible only by the prime numbers in $\pi$ while the index is coprime to any number in $\pi$), and all Hall $\pi$-subgroups are conjugate. A finite solvable group has a Hall $\pi$-subgroup for any set $\pi$ of prime numbers. Every $\pi$-subgroup of a finite solvable group is contained in a Hall $\pi$-subgroup, and all Hall $\pi$-subgroups are conjugate. A normal Hall subgroup $H$ of a finite group $G$ always has a complement in $G$, that is, a subgroup $D$ such that $G=H\cdot D$ and such that $H\cap D$ is trivial; all complements to $H$ in $G$ are conjugate. If a group has a nilpotent Hall $\pi$-subgroup (cf. Nilpotent group), then all Hall $\pi$-subgroups are conjugate, and every $\pi$-subgroup is contained in some Hall $\pi$-subgroup. In general, a Hall subgroup does not have these properties. For example, the alternating group $A_5$ of order 60 has no Hall $\{2,5\}$-subgroup. In $A_5$ there is a Hall $\{2,3\}$-subgroup of order 12, but there is a subgroup of order 6 which does not lie in a Hall subgroup. Finally, in the simple group of order 168 the Hall $\{2,3\}$-subgroups are not conjugate.

References

[1]	S.A. Chunikhin, "Subgroups of finite groups" , Wolters-Noordhoff (1969) (Translated from Russian)
[2]	Itogi Nauk. i Tekhn. Algebra. 1964 (1966) pp. 7–46
[3]	B. Huppert, "Endliche Gruppen" , 1 , Springer (1979) pp. 482–490
[4]	D. Gorenstein (ed.) , Reviews on finite groups , Amer. Math. Soc. (1974)