PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A split extension is a direct product exactly when its complement centralizes the kernel

Statement

Let

$1 \to N \overset{i}{\to} E \overset{π}{\to} Q \to 1$

be a split group extension, and let $C \leq E$ be a complement to $i (N)$ . Then $E$ is the internal direct product of $i (N)$ and $C$ if and only if every element of $C$ commutes with every element of $i (N)$ .

Facts & Assumptions

Given: The displayed split extension and a complement $C \leq E$ .

[L1]

The complement $C$ induces an action of $Q$ on $N$ by conjugation (A complement determines the conjugation action on the kernel).

[L2]

In a semidirect product, the internal decomposition is direct exactly when the action is trivial (The canonical semidirect decomposition is an internal direct product if and only if the defining action is trivial).

Proof

technique · iff

1.1

By [L1], the split extension is equivalent to the semidirect-product extension defined by the complement-induced action of $Q$ on $N$ .

givenL1

2.1

If every element of $C$ commutes with every element of $i (N)$ , then each conjugation automorphism from step 1.1 is the identity. So the induced action is trivial, and [L2] makes the decomposition a direct product.

L1L2step 1.1

3.1

Conversely, if the decomposition is a direct product, then the induced action is trivial by [L2]. Therefore the conjugation of $i (N)$ by every element of $C$ is the identity, so $C$ centralizes $i (N)$ .

L1L2step 1.1∎

Depends on

Used by

The split extension C₂ × C₂ of C₂ by C₂ is direct Example
FALSE: every split group extension is a direct product False statement

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Group Theory (standard reference, not scraped)