Split extension

A split extension is a group extension that can be reconstructed completely from an action of the quotient on the kernel. Splitting is equivalent both to the existence of a complement and to a homomorphic section of the quotient map; the resulting groups are precisely semidirect products. The subject converts extension questions into automorphism actions, conjugacy of complements, and crossed homomorphisms. ^{[@isaacs2008]}

Group extensions

The abstract isomorphism types of $$N$$ and $$Q$$ do not determine $$G$$ . Both $$C_4$$ and $$V_4$$ are extensions of $$C_2$$ by $$C_2$$ . One must also know the induced action on the kernel and, in general, twisting data.

Equivalent splitting conditions

External semidirect products

Given groups $$N,H$$ and a homomorphism

\varphi:H\longrightarrow\operatorname{Aut}(N),

define multiplication on $N\times H$ by

(n,h)(m,k)=\bigl(n\varphi(h)(m),hk\bigr).

The resulting group is denoted $N\rtimes_\varphi H$ .

When $\varphi$ is trivial, the construction is the direct product $N\times H$ . The semidirect product records precisely the nontrivial conjugation action of a complement on the normal kernel.

Internal semidirect products

If $N,H\leq G$ satisfy

N\triangleleft G, \qquad N\cap H=1, \qquad G=NH,

then $$G$$ is the internal semidirect product of $$N$$ by $$H$$ . Every element has a unique expression $$nh$$ . If $$H$$ is also normal, then $[N,H]\leq N\cap H=1$ , and the semidirect product is actually direct.

Actions and isomorphism classification

Suppose two actions $\varphi,\psi:H\to\operatorname{Aut}(N)$ satisfy

\psi(\beta(h))=\alpha\varphi(h)\alpha^{-1}

for some $\alpha\in\operatorname{Aut}(N)$ and $\beta\in\operatorname{Aut}(H)$ . Then their semidirect products are isomorphic. Conversely this orbit condition is necessary when the isomorphism is required to preserve the designated kernel and quotient.

For $$N=C_p$$ ,

\operatorname{Aut}(C_p)\cong C_{p-1}.

The existence of a nonabelian group of order $$pq$$ with $$q<p$$ is therefore controlled by a nontrivial map $C_q\to C_{p-1}$ , equivalently by $q\mid p-1$ . This explains the classification first encountered through Sylow counting.

Complements and crossed homomorphisms

Fix $G=N\rtimes H$ . Another complement can often be written as the graph of a function $\delta:H\to N$ :

H_\delta=\{(\delta(h),h)\mid h\in H\}.

It is a subgroup exactly when

\delta(hk)=\delta(h)\,{}^h\!\delta(k),

so $\delta$ is a crossed homomorphism, or nonabelian $$1$$ -cocycle. Conjugating complements by elements of $$N$$ corresponds to principal crossed homomorphisms. Thus $$N$$ -conjugacy classes of complements are governed by the nonabelian cohomology set $$H^1(H,N)$$ .

This says that finite coprime extensions split and that their complements are essentially unique under broad hypotheses. See Schur–Zassenhaus theorem for the proof.

Standard examples

Nonsplit extensions

These examples show that splitting is not determined by the abstract kernel and quotient alone. For an abelian kernel, the obstruction is the extension class in $$H^2(Q,N)$$ .

Relation with wreath products

The Kaloujnine–Krasner theorem embeds every extension in a suitable wreath product. A split extension is the most direct case: the action $H\to\operatorname{Aut}(N)$ already determines multiplication, so no nontrivial $$2$$ -cocycle is required. Isaacs, Chapter 3 uses wreath products to organize conjugation actions and coprime extensions. ^{[@isaacs2008]}

This article follows the complement-and-action line of Isaacs, Chapter 3 and adds the standard internal/external construction, cohomological obstruction, and nonsplit examples. Terminology was cross-checked against the cited public encyclopedia revisions. ^{[@isaacs2008]} ^{[@robinson1996]} ^{[@wikipedia-semidirect]} ^{[@wikipedia-semidirect-en]}