Hall subgroup

A Hall subgroup is a subgroup whose order is coprime to its index. Hall subgroups generalize Sylow subgroups from a single prime to a set of primes, and they are the natural setting for splitting a finite group as a semidirect product.

Definition

A subgroup H ≤ G of a finite group is a Hall subgroup if

gcd(|H|, [G : H]) = 1,

i.e. |H| and its index [G : H] = |G| / |H| share no common prime.

Fix a set of primes π. A positive integer is a π-number if every prime dividing it lies in π. A Hall π-subgroup is a subgroup whose order is a π-number and whose index is divisible by no prime in π — equivalently, a Hall subgroup H with π the set of primes dividing |H|.

Intuition

Write |G| as (π-part) · (π′-part). A Hall π-subgroup captures the entire π-part of the order in a single subgroup. Sylow's theorem is the case π = {p}: a Sylow p-subgroup is a Hall {p}-subgroup, taking the full power of one prime. Sylow subgroups always exist; Hall subgroups for larger π need not — their existence is tied to solvability.

Examples

• Sylow subgroups. Every Sylow p-subgroup is a Hall {p}-subgroup.
• Trivial cases. {e} (the Hall ∅-subgroup) and G itself (the Hall π-subgroup for π = all primes dividing |G|).
• S₃ ≅ C₃ ⋊ C₂. The order-3 subgroup A₃ is a normal Hall {3}-subgroup; the three order-2 subgroups are its complements, exhibiting S₃ as the semidirect product C₃ ⋊ C₂.
• S₄ (order 24 = 2³·3). A Hall {2}-subgroup has order 8 (a Sylow 2-subgroup, ≅ D₄); a Hall {3}-subgroup has order 3; the Hall {2,3}-subgroup is S₄ itself.
• A₅ (order 60 = 2²·3·5). A₄ ≤ A₅ has order 12 and index 5, a Hall {2,3}-subgroup.

Non-examples

• In A₅ there is no subgroup of order 15, so A₅ has no Hall {3,5}-subgroup. Existence can fail because A₅ is not solvable.
• In S₄, ⟨(12)⟩ is not a Hall subgroup: |H| = 2 and [G : H] = 12 share the prime 2.

Properties

• A Sylow p-subgroup is a Hall subgroup for the single prime p. Hall subgroups always exist for one prime; for larger sets of primes they may not.
• Hall subgroups can fail to exist: A₅ (order 2²·3·5) has no subgroup of order 15, so no Hall {3,5}-subgroup — possessing all of them requires solvability.
• Hall's theorem: a finite solvable group has a Hall π-subgroup for every set of primes π, all of them conjugate, and every π-subgroup lies inside one.
• Hall subgroups detect solvability: if G has a Hall p′-subgroup for every prime p, then G is solvable.
• Schur–Zassenhaus: a normal Hall subgroup N ⊴ G always has a complement, so G = N ⋊ K is a semidirect product with |K| = [G : N], and the complements are all conjugate.
• A normal Hall subgroup is unique and characteristic: it is the only subgroup of its order.

See also

• Sylow subgroup — the single-prime case, π = {p}.
• Normal subgroup — a normal Hall subgroup is the unique subgroup of its order, and is complemented.
• Semidirect product — Schur–Zassenhaus splits a normal Hall subgroup off as a complemented factor.
• Conjugacy class — Hall π-subgroups of a solvable group form a single conjugacy class.
• Solvable group — existence and conjugacy of all Hall subgroups characterizes solvability.
• Wikipedia: Hall subgroup