Group extension

A group extension is a group G together with a normal subgroup N and the quotient Q = G / N. Every finite group is an iterated extension of simpler groups; describing all extensions of a given Q by a given N is one of the core classification problems.

Definition

A short exact sequence of groups is a sequence

1 ⟶ N ⟶ G ⟶ Q ⟶ 1

in which the left map is injective, the right map is surjective, and the image of the left equals the kernel of the right. Equivalently: G has a normal subgroup isomorphic to N with quotient isomorphic to Q. We call G an extension of Q by N.

Conventions differ. Some authors call G an extension of N by Q. Reading carefully is always necessary; we'll say "G fits in the sequence above" when clarity is needed.

Split versus non-split

The extension splits if there is a homomorphism s : Q → G with π ∘ s = idQ, where π : G → Q is the quotient map. A split extension is the same thing as a semidirect product: G ≅ N ⋊ Q, where Q acts on N by conjugation through s.

A non-split extension needs genuinely new information beyond N and Q and the action.

Central extensions

An extension is central if N ⊆ Z(G). Central extensions of Q by an abelian N are classified by the second cohomology group H²(Q, N), with the trivial class corresponding to the direct product N × Q.

The cohomological classification is the engine behind the Besche-Eick-O'Brien enumeration — extensions of a given quotient by a given elementary abelian p-group correspond to elements of H² with the right action.

Examples

• Q₈ is a non-split central extension 1 → C₂ → Q₈ → V → 1. There's no section: every element of Q₈ ∖ {±1} has order 4, while every element of V has order ≤ 2.
• D₄ fits into the same diagram 1 → C₂ → D₄ → V → 1 — but this one splits (take s : V → D₄ sending the two V-generators to r² and s). Both D₄ and Q₈ are extensions with the same (N, Q) pair; they differ in the H² class.
• S₄ extends S₃ by V₄: 1 → V₄ → S₄ → S₃ → 1. Splits (pick a copy of S₃ ≤ S₄ fixing a point).
• SL(2,𝔽3) is a non-split extension 1 → C₂ → SL(2,𝔽3) → A₄ → 1.

Extension problem

Given N and Q, classify (up to isomorphism) all groups G fitting in 1 → N → G → Q → 1.

• Abelian N with a given Q-action. Classified by H²(Q, N).
• Non-abelian N. Classified by a pair consisting of a group homomorphism Q → Out(N) plus an obstruction in H³(Q, Z(N)); when this obstruction vanishes the extensions are in bijection with H²(Q, Z(N)).

See also

• Semidirect product — split extensions.
• Direct product — trivial central extension.
• SmallGroups enumeration — uses H² to enumerate extensions by elementary abelian p-groups.
• Simple group — endpoints of the iterated-extension story.
• Wikipedia: Group extension