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Proof By Proposition 3 and as a result of NFT, the simplest (unique reduced) characteristic polynomial in system (5) represents the congruence class of the related l-classes.
Example 2 Let p = 3 and M = [ 1 0 0 2 ] with determinant 2. Thus M 7 ≡ M mod 3 holds, and the relators of the semi-direct product presentation, as in (4), will be y 1 y 2 = y 2 y 1, x 7 = x, y 1 x = x y 1 and y 2 x = x y 2 2. By Theorem 3, the simplest (unique) characteristic polynomial ν 2 − 3 ν + 2 represents this inefficient but minimal presentation.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com