Root-of-unity representatives of finite-group factor sets

Over an algebraically closed field, every normalized factor set of a finite group can be rescaled to take values in the roots of unity of order dividing the group order. This gives a finite set of representatives for the cohomology classes of factor sets, and hence finiteness of the Schur multiplier.

The explicit rescaling comes from taking roots of the products of the rows of the factor set. It works in arbitrary characteristic and does not require a faithful projective representation.

References

• G. Karpilovsky, Projective Representations of Finite Groups (1985), Chapter 2.

theorem TauCeti.IsFactorSet.exists_rescale_pow_card_eq_one {k : Type u_1} {G : Type u_2} [Field k] [IsAlgClosed k] [Group G] [Finite G] (α : G → G → kˣ) [IsFactorSet α] : ∃ (c : G → kˣ), c 1 = 1 ∧ ∀ (g h : G), (c g * c h * (c (g * h))⁻¹ * α g h) ^ Nat.card G = 1

A normalized factor set of a finite group over an algebraically closed field can be rescaled by a normalized cochain so that every value has order dividing Nat.card G.