We propose a classification of finite local nearrings with multiplicative Shmidt group. Moreover, it is shown that there are no nearrings with identity on the Shmidt groups.
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H. Zassenhaus, “Über endliche Fastkörper,” Abh. Math. Semin. Univ. Hamburg, 11, 187–220 (1935/36).
S. Ligh, “Finite hereditary near-field groups,” Monatsh. Math., 86, 7–11 (1978).
M. Yu. Raevs’ka, “Local nearrings with multiplicative Miller–Moreno group,” Visn. Kyiv. Univ., Mat. Mekh., 25, 45–48 (2011).
A. Gorodnik, “Local nearrings with commutative groups of units,” Houston J. Math., 25, 223–234 (1999).
B. Amberg, P. Hubert, and Ya. Sysak, “Local nearrings with dihedral multiplicative group,” J. Algebra, 273, 700–717 (2004).
P. Hubert, Nearrings and a Construction of Triply Factorized Groups, Ph.D. Thesis, Universität Mainz (2005).
Ya. P. Sysak and Di S. Termini, “Local nearrings with generalized quaternion multiplicative group,” Ric. Mat., 56, 61–72 (2007).
M. Yu. Raevs’ka and Ya. P. Sysak, “On local near-rings with Miller–Moreno multiplicative group,” Ukr. Mat. Zh., 64, No. 6, 811–818 (2012); English translation:Ukr. Math. J., 64, No. 6, 930–937 (2012).
J. R. Clay and Jr. Malone, “The near-rings with identities on certain finite groups,” Math. Scand., 19, 146–150 (1966).
J. R. Clay and D. Doi, “Near-rings with identity on alternating groups,” Math. Scand., 23, 54–56 (1968).
J. R. Clay, “Research in near-ring theory using a digital computer,” BIT, 10, 249–265 (1970).
T. H. H. Boykett and C. Nobauer, “A class of groups which cannot be the additive groups of near-rings with identity,” in: Proc. of the Klagenfurt Conference “Contributions to General Algebra”(Klagenfurt, 1997), Heyn, Klagenfurt (1998), pp. 89–99.
I. Yu. Raievska and M. Yu. Raievska, “Finite nearrings with identity on Miller Moreno groups,” Mat. Stud., 42, No. 1, 15–20 (2014).
I. Yu. Raievska and Ya. P. Sysak, “Finite local nearrings on metacyclic Miller Moreno p-groups,” Algebra Discrete Math., 13, No. 1, 111–127 (2012).
C. J. Maxson, “Local near-rings of cardinality p2,” Canad. Math. Bull., 11, No. 4, 555–561 (1968).
I. Yu. Raevs’ka and M. Yu. Raevs’ka, “Local nearrings with restrictions imposed on the multiplicative groups and subgroups of noninvertible elements,” Vauk. Chasopys. Drahomanov Nats. Ped. Univ., Ser. 1, Fiz.-Mat. Nauk., 14, 134–145 (2013).
O. Yu. Shmidt, “Groups all subgroups of which are special,” Mat. Sb., 31, 366–372 (1924).
B. Huppert, Endliche Gruppen I, Springer, Berlin (1967).
D. Gorenstein, Finite Groups, Harper & Row, New York (1968).
The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.8.8 (2017); https://www.gap-system.org.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 10, pp. 1435–1440, October, 2019.
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Raievska, I.Y., Raievska, M.Y. Local Nearrings with Multiplicative Shmidt Group. Ukr Math J 71, 1643–1649 (2020). https://doi.org/10.1007/s11253-020-01737-6
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DOI: https://doi.org/10.1007/s11253-020-01737-6