Group extensions and cohomology

Extensions and complements (Schur–Zassenhaus), splitting of extensions, homology and first cohomology of groups, cohomological dimension, and integrability of groups as derived subgroups.

• Cohomological dimension of just-non-polycyclic groups
• Dimension of first cohomology groups
• Extensions of the additive group of a division ring
• Homology of finitely generated metabelian groups
• Integrable groups
• Ranks in elementary abelian extensions
• Self-extensions of groups
• Surjections injective on no nontrivial subgroup

TheoremSchur–Zassenhaus theorem: conjugacy of complements#✓fully proved

Let \(G\) be a finite group and \(N\) a normal subgroup of \(G\) whose order is coprime to its index, and suppose \(N\) is solvable. If \(K_1\) and \(K_2\) are complements of \(N\) in \(G\), then there is an element \(g \in N\) with \(g K_1 g^{-1} = K_2\).

theorem schur_zassenhaus_conjugate_complements
    {G : Type*} [Group G] [Finite G]
    (N : Subgroup G) [N.Normal]
    (hN : Nat.Coprime (Nat.card N) N.index)
    [Group.IsSolvable N]
    (K₁ K₂ : Subgroup G)
    (h₁ : N.IsComplement' K₁) (h₂ : N.IsComplement' K₂) :
    ∃ g : G, g ∈ N ∧ K₁.map (MulAut.conj g).toMonoidHom = K₂

Sources

• I. Schur (1904); H. Zassenhaus, Lehrbuch der Gruppentheorie (1937); Isaacs, Finite Group Theory; standard solvability hypothesis for conjugacy.

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