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

  1. Hall subgroups exist in finite solvable
  2. Hall implies order-conjugate in finite solvable
  3. Hall implies order-dominating in finite solvable
  4. Congruence condition on factorization of Hall numbers

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