D-property for a prime set

Definition

Let $G$ be a finite group, and $\pi$ be a set of primes. We say that $G$ satisfies $D_{\pi }$ if $D$ has a $\pi$-Hall subgroup $H$ (viz, a Hall subgroup whose order contains only primes from $\pi$ and whose index contains no primes from $\pi$) and every $\pi$-subgroup (subgroup whose order contains only primes from $\pi$) is contained in a conjugate of $H$.

The D here stands for Domination and relates to the problem of whether we can extend Sylow's theorems to Hall subgroups.

Relation with other properties

Weaker properties

• C-property for a prime set
• E-property for a prime set