ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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A shear and a quarter-turn exhibit noncommuting automorphisms of $Z^{2}$

Example

The integer arrays

$A = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}), B = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$

define automorphisms of $Z^{2}$ that do not commute.

Facts & Assumptions

Given: The two displayed integer arrays.

[L1]

Automorphisms of $Z^{2}$ correspond to invertible two-by-two integer arrays, and composition corresponds to array multiplication ( $Aut (Z^{n}) ≅ G L_{n} (Z)$ for every finite rank $n$ ).

Verification

technique · direct

1.1L1algebra

Integer inverses are $A^{- 1} = (\begin{array}{cc} 1 & - 1 \\ 0 & 1 \end{array})$ and $B^{- 1} = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ . Thus both arrays define automorphisms by [L1].

2.1step 1.1L1algebra∎

Direct multiplication gives $A B = (\begin{array}{cc} 1 & - 1 \\ 1 & 0 \end{array})$ and $B A = (\begin{array}{cc} 0 & - 1 \\ 1 & 1 \end{array})$ . They differ, so the corresponding automorphisms do not commute.

Depends on

$\operatorname{Aut}(\mathbb Z^n)\cong GL_n(\mathbb Z)$ for every finite rank $n$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.

A shear and a quarter-turn exhibit noncommuting automorphisms of Z2Z2