Geodesic
Classes of piecewise-quasiaffine transformations on the generalized 2-group of quaternions
Diskretnaya Matematika, Tome 34 (2022) no. 1, pp. 103-125

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The class of nonabelian 2-groups $H$ with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group $H$, and put forward criteria of their bijectivity. For the generalized group of quaternions of order $2^m$, we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities.
Keywords: orthomorphism, complete transformation, dihedral group, generalized quaternion group, semidihedral group, modular maximal cyclic group. 
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     author = {B. A. Pogorelov and M. A. Pudovkina},
     title = {Classes of  piecewise-quasiaffine transformations on the generalized 2-group of quaternions},
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     pages = {103--125},
     publisher = {mathdoc},
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     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2022_34_1_a7/}
}
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B. A. Pogorelov; M. A. Pudovkina. Classes of  piecewise-quasiaffine transformations on the generalized 2-group of quaternions. Diskretnaya Matematika, Tome 34 (2022) no. 1, pp. 103-125. http://geodesic.mathdoc.fr/item/DM_2022_34_1_a7/