# ECD condition for pi-subgroups in solvable groups

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## Statement

In a finite solvable group, the set of $\pi$-subgroups, for any prime set $\pi$ (i.e., the set of subgroups that satisfy the condition that all prime divisors of the order are in $\pi$), satisfies the ECD condition with the maximal elements being Hall $\pi$-subgroups. Explicitly, for any $\pi$:

• Existence (E): There exists a Hall $\pi$-subgroup
• Conjugacy (C): Any two Hall $\pi$-subgroups are conjugate
• Domination (D): Any $\pi$-subgroup is contained in a Hall $\pi$-subgroup. Equivalently, given any particular Hall $\pi$-subgroup, every $\pi$-subgroup is conjugate to a subgroup contained within this Hall $\pi$-subgroup
• Number (N): The number of Hall $\pi$-subgroups is a product of factors, each of which is congruent to $1$ modulo some $p \in \pi$.

It turns out that conversely, if Hall $\pi$-subgroups exist for every prime set $\pi$ and a given finite group, then the finite group is solvable. This nontrivial result is termed Hall's theorem, and relies on the hard Burnside's p^aq^b theorem, which proves it for the case where there are only two prime divisors of the order.

## Proof components

Also related is: Divisibility condition on factorization of Hall numbers.