ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-05

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The C_p^2 extension as a nonzero two-cocycle

Example

For the nonsplit extension

$0 \to C_{p} \to C_{p^{2}} \to C_{p} \to 0$

with trivial action on the kernel, a normalized section yields the cocycle

$f (i, j) = {\begin{cases} 0, & i + j < p, \\ 1, & i + j \geq p, \end{cases}$

for $0 \leq i, j < p$ . This cocycle is nonzero in $H^{2} (C_{p}, C_{p})$ .

Facts & Assumptions

Given: The quotient map $Z / p^{2} Z \to Z / p Z$ and the section $s (i) = i$ for $0 \leq i < p$ .

[L1]

An extension determines a well-defined class in $H^{2}$ (An extension determines a well-defined H^2 class).

[L2]

$H^{2}$ classifies extensions with fixed abelian kernel action (H^2 classifies extensions with fixed abelian kernel action).

Verification

technique · direct

1.1

In $Z / p^{2} Z$ , adding two chosen lifts $i$ and $j$ either stays below $p$ or crosses the first multiple of $p$ . Therefore $s (i) + s (j) - s (i + j m o d p) = p f (i, j),$ with $f (i, j)$ given by the carry function above.

givenalgebra

2.1

The kernel is central, so [L1] identifies this carry function with the extension class. If it were a coboundary, then [L2] would make the extension split, but $C_{p^{2}}$ has no subgroup of order $p$ complementary to its unique subgroup of order $p$ .

L1L2step 1.1algebra

3.1

Hence the displayed cocycle represents a nonzero class in $H^{2} (C_{p}, C_{p})$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)