Geodesic
On affine classification of permutations on the space $GF(2)^3$
Diskretnaya Matematika, Tome 30 (2018) no. 3, pp. 77-87

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We give an elementary proof that by multiplication on left and right by affine permutations $A,B\in AGL(3,2)$ each permutation $\pi:GF(2)^3\rightarrow GF(2)^3$ may be reduced to one of the 4 permutations for which the $3\times3$-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory.
Keywords: permutation, affine transformation, Pólya theory, de Brouijn's theorem. 
@article{DM_2018_30_3_a6,
     author = {F. M. Malyshev},
     title = {On affine classification of permutations on the space $GF(2)^3$},
     journal = {Diskretnaya Matematika},
     pages = {77--87},
     publisher = {mathdoc},
     volume = {30},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2018_30_3_a6/}
}
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F. M. Malyshev. On affine classification of permutations on the space $GF(2)^3$. Diskretnaya Matematika, Tome 30 (2018) no. 3, pp. 77-87. http://geodesic.mathdoc.fr/item/DM_2018_30_3_a6/