CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 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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Extensions with coprime kernel and quotient split

Statement

Let

$1 \to N \to G \to Q \to 1$

be an extension of finite groups. If $\gcd (∣ N ∣, ∣ Q ∣) = 1$ , then the extension splits.

Facts & Assumptions

Given: The displayed extension of finite groups, with $\gcd (∣ N ∣, ∣ Q ∣) = 1$ .

[L1]

In a group extension, the kernel is normal and the quotient recovers the base (In a group extension the kernel is normal and the quotient recovers the base).

[L2]

A normal Hall subgroup has a complement (Schur-Zassenhaus existence theorem).

[L3]

A complement to the kernel is equivalent to a splitting section (A group extension splits exactly when it has a complement or a compatible semidirect-product model, and a kernel retraction forces a direct product).

Proof

technique · direct

1.1givenL1algebra

By [L1], the kernel $N$ is normal in $G$ and $G / N ≅ Q$ , so $[G : N] = ∣ G / N ∣ = ∣ Q ∣$ . Because $\gcd (∣ N ∣, ∣ Q ∣) = 1$ , no prime divisor of $[G : N]$ divides $∣ N ∣$ . Hence $N$ is a Hall subgroup of $G$ .

2.1L2L3step 1.1∎

By [L2], the normal Hall subgroup $N$ has a complement in $G$ . Then [L3] turns that complement into a splitting section, so the extension splits.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)