CounterexampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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$1 \to C_{p} \to C_{p^{2}} \to C_{p} \to 1$ does not split

Statement refuted

Every short exact sequence of groups splits.

For every prime $p$ , the sequence

$1 ⟶ C_{p} ⟶ C_{p^{2}} ⟶ C_{p} ⟶ 1$

obtained from multiplication by $p$ and reduction modulo $p$ is a counterexample.

Facts & Assumptions

Given: A prime $p$ , written additively with the middle group $Z / p^{2}$ .

[L1]

A split extension has a homomorphic section of its quotient map (Group extensions, sections, complements, and split extensions).

[L2]

Every subgroup of a cyclic group is cyclic (Every subgroup of a cyclic group is cyclic; the least positive exponent in a nontrivial subgroup supplies a generator).

[L3]

An element $x$ of finite order $r$ satisfies $m x = 0$ exactly when $r ∣ m$ (If $ord (g) = n$ then $g^{k} = e$ iff $k$ is an integer multiple of $n$ , the powers $g^{0}, \dots, g^{n - 1}$ are distinct, and $⟨ g ⟩$ has exactly $n$ elements; if $g$ has infinite order then $g^{j} = g^{k}$ only for $j = k$ ).

Counterexample

technique · contradiction

1.1L3algebra

The injection sends $[a]_{p}$ to $[p a]_{p^{2}}$ , and the quotient map $π$ sends $[x]_{p^{2}}$ to $[x]_{p}$ . The kernel of $π$ is the subgroup $p C_{p^{2}}$ of order $p$ , so the sequence is short exact.

1.2L1L2L3assume-contraalgebra

Suppose, for contradiction, that a section $s : C_{p} \to C_{p^{2}}$ existed. Since $π s$ is the identity, $s$ is injective, so its image $H$ is a subgroup of order $p$ . By [L2], $H = ⟨ [a]_{p^{2}} ⟩$ for some $a$ . Its generator has order $p$ , so [L3] gives $p^{2} ∣ p a$ and hence $p ∣ a$ . Thus $H \subseteq p C_{p^{2}}$ ; both subgroups have $p$ elements, so they are equal.

2.1step 1.2L1discharge-contradiction∎

Then $π s$ is the zero homomorphism, contradicting that it is the identity on the nontrivial group $C_{p}$ . Thus the extension does not split.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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)

1→Cp→Cp2→Cp→11→Cp​→Cp2​→Cp​→1 does not split