Local nearrings, their structure, and the GAP system
Abstract
UDC 512.6
We present an overview of the current state of research of finite local nearrings. In particular, it contains all recently obtained results on the classification of local nearrings.
References
E. Aichinger, F. Binder, Ju. Ecker, P. Mayr, C. Noebauer, SONATA --- System of Nearrings and their Applications, Version 2.9.6, Johannes Kepler Universitaet Linz (2022); (https://gap-packages.github.io/sonata/.
B. N. Al-Hasanat, A. Almazaydeh, On classification of groups of order $p^4,$ where $p$ is an odd prime, Int. J. Math. and Comput. Sci., 17, № 4, 1569–1593 (2022).
B. Amberg, P. Hubert, Ya. Sysak, Local near-rings with dihedral multiplicative group, J. Algebra, 273, 700–717 (2004). DOI: https://doi.org/10.1016/j.jalgebra.2003.10.007
M. Arunma, D. Ezhilmaran, A new multiparty identification scheme based on nearring root extraction problem, J. Discrete Math. Sci. and Cryptogr., 22, № 1, 59–64 (2019). DOI: https://doi.org/10.1080/09720529.2019.1569820
W. Burnside, Theory of groups of finite order, Univ. Press, Cambridge (1897).
T. H. H. Boykett, C. Nobauer, A class of groups which cannot be the additive groups of near-rings with identity, Contributions to General Algebra, 10 (Klagenfurt, 1997), Heyn, Klagenfurt, 89–99 (1998).
J. R. Clay, Research in near-ring theory using a digital computer, BIT, 10, 249–265 (1970). DOI: https://doi.org/10.1007/BF01934196
J. R. Clay, Nearrings. Geneses and applications, Clarendon Press, Oxford Univ. Press, New York (1992). DOI: https://doi.org/10.1093/oso/9780198533986.001.0001
J. R. Clay, D. Doi, Near-rings with identity on alternating groups, Math. Scand., 23, 54–56 (1968). DOI: https://doi.org/10.7146/math.scand.a-10896
J. R. Clay, Jr. Malone, The near-rings with identities on certain finite groups, Math. Scand., 19, 146–150 (1966). DOI: https://doi.org/10.7146/math.scand.a-10803
J. R. Clay, C. J. Maxson, The near-rings with identities on generalized quaternion groups, Ist. Lombardo Accad. Sci. Lett. Rend. A, 104, 525–530 (1970).
L. Dixon, Definitions of a group and a field by independent postulates, Trans. Amer. Math. Soc., 6, 198–204 (1905). DOI: https://doi.org/10.1090/S0002-9947-1905-1500706-2
D. Ezhilmaran, V. Muthukumaran, Key exchange protocol using decomposition problem in near-ring, Gazi Univ. J. Sci., 29, № 1, 123–127 (2016).
D. Ezhilmaran, V. Muthukumaran, Authenticated group key agreement protocol based on twist conjugacy problem in near-rings, Wuhan Univ. J. Natural Sci., 22, № 6, 472–476 (2017). DOI: https://doi.org/10.1007/s11859-017-1275-9
S. Feigelstock, Additive groups of local near-rings, Comm. Algebra, 34, 743–747 (2006). DOI: https://doi.org/10.1080/00927870500388042
C. C. Ferrero, G. Ferrero, Nearrings. Some developments linked to semigroups and groups, Kluwer Acad. Publ., Dordrecht (2002). DOI: https://doi.org/10.1007/978-1-4613-0267-4
P. Furtwangler, O. Taussky-Todd, Über Schiefringe, Sitzber. Akad. Wiss. Wien, Math. Nat. Klasse, Abt. IA, 145, 525 (1936).
The GAP Group, Aechen, St Andrews, GAP --- Groups, Algorithms and Programming, Version 4.13.0 (2024); https://www.gap-system.org/.
A. Gorodnik, Local near-rings with commutative groups of units, Houston J. Math., 25, 223–234 (1999).
M. Jr. Hall, The theory of groups, MacMillan Company, New York (1959).
P. Hubert, Nearrings and a construction of triply factorized groups, Mainz (2005).
M. J. Johnson, Near-rings with identities on dihedral groups, Proc. Edinb. Math. Soc. (2), 18, 219–228 (1972/73). DOI: https://doi.org/10.1017/S0013091500009950
L. A. Kaluzhnin, V. I. Suhchanskyy, Verbal functions on groups, Theoretical and Applied Questions of Differential Equations and Algebra (in Russian), 264, Naukova Dumka, Kiev (1978), p. 105–110.
B. W. King, Presentations of metacyclic groups, Bul. Austral. Math. Soc., 8, 103–131 (1973). DOI: https://doi.org/10.1017/S0004972700045500
J. E. Krimmel, A condition on near-rings with identity, Monatsh. Math., 77, 52–54 (1973). DOI: https://doi.org/10.1007/BF01300528
R. R. Laxton, R. Lockhart, The near-rings hosted by a class of groups, Proc. Edinb. Math. Soc., 23, 69–86 (1980). DOI: https://doi.org/10.1017/S001309150000362X
S. Ligh, Near rings with identities on certain groups, Monatsh. Math., 75, 38–43 (1971). DOI: https://doi.org/10.1007/BF01305976
R. Lockhart, The theory of near-rings, Lecture Notes in Math., 2295, Springer, Cham (2021). DOI: https://doi.org/10.1007/978-3-030-81755-8
C. J. Maxon, On local near-rings, Math. Z., 106, 197–205 (1968). DOI: https://doi.org/10.1007/BF01110133
C. J. Maxon, Local near-rings of cardinality $p^2$, Canad. Math. Bull., 11, № 4, 555–561 (1968). DOI: https://doi.org/10.4153/CMB-1968-066-2
C. J. Maxon, On the construction of finite local near-rings (I): on non-cyclic Abelian $p$-groups, Quart. J. Math. Oxford (2), 21, 449–457 (1970). DOI: https://doi.org/10.1093/qmath/21.4.449
C. J. Maxon, On the construction of finite local near-rings (II): on non-Abelian $p$-groups, Quart. J. Math. Oxford (2), 22, 65–72 (1971). DOI: https://doi.org/10.1093/qmath/22.1.65
J. D. P. Meldrum, Near-rings and their links with groups, PPL (1985).
F. Menegazzo, Automorphisms of $p$-groups with cyclic commutator subgroup, Rend. Semin. Mat. Univ. Padova, 90, 81–101 (1993).
R. J. Miech, On $p$-groups with a cyclic commutator, J. Austral. Math. Soc., 20, 178–198 (1975). DOI: https://doi.org/10.1017/S1446788700020486
G. A. Miller, H. Moreno, Non-Abelian groups in which every subgroup is Abelian, Trans. Amer. Math. Soc., 4, 398–404 (1903). DOI: https://doi.org/10.1090/S0002-9947-1903-1500650-9
V. Muthukumaran, D. Ezhilmaran, I. Muchtadi-Alamsyah, R. Udhayakumar, A. Manickam, New public key cryptosystem based on combination of NREP and CSP in noncommutative near-ring, J. Xi'an Univ. Architecture and Technology, 12, Issue III, 4534–4540 (2020).
Q. Ore, Linear equations in non-commutative fields, Ann. Math., 32, 463–477 (1931). DOI: https://doi.org/10.2307/1968245
G. Pilz, Near-rings. The theory and its applications, second ed., North-Holland, Amsterdam (1983).
I. Yu. Raievska, Local nearrings with additive groups of order 128, Наук. вісн. Ужгород. університету. Математика і інформатика (2024).
I. Yu. Raievska, M. Yu. Raievska, Finite nearrings with identity on Miller–Moreno groups, Mat. Stud., 42, № 1, 15–20 (2014).
I. Yu. Raievska, M. Yu. Raievska, Local nearrings on finite non-Abelian $2$-generated $p$-groups, Carpathian Math. Publ., 12, № 1, 199–207 (2020). DOI: https://doi.org/10.15330/cmp.12.1.199-207
I. Raievska, M. Raievska, Lower bounds for the number of local nearrings on groups of order $p^3$, Math. Commun., 29, 177–-191 (2024).
I. Raievska, M. Raievska, Groups of the nilpotency class $2$ of order $p^4$ as additive groups of local nearrings, Algebra and Discrete Math., 38, № 1, 93–114 (2024). DOI: https://doi.org/10.12958/adm2269
I. Raievska, M. Raievska, Groups of the nilpotency class $3$ of order $p^4$ as additive groups of local nearrings, Carpathian Math. Publ. (2024); https://arxiv.org/abs/2309.14342. DOI: https://doi.org/10.1007/s11253-024-02369-w
I. Yu. Raievska, Ya. P. Sysak, Finite local nearrings on metacyclic Miller–Moreno $p$-groups, Algebra and Discrete Math., 13, № 1, 111–127 (2012).
I. Yu. Raievska, M. Yu. Raievska, Ya. P. Sysak, On local nearrings of order 32, International Conference dedicated to 100th anniversary of S. M. Chernikov (August 20–26, 2012, Kyiv, Ukraine), Book of abstracts, Institute of Mathematics of NAS of Ukraine, Kyiv (2012), p. 124.
I. Yu. Raievska, M. Yu. Raievska, Ya. P. Sysak, Finite local nearrings with split metacyclic additive group, Algebra and Discrete Math., 22, № 1, 129–152 (2016).
I. Yu. Raievska, M. Yu. Raievska, Ya. P. Sysak, Local nearrings of order 64, International Mathematical Conference ``Groups and Actions: Geometry and Dynamics, dedicated to the memory of professor Vitaly Sushchanskyy'' (December 19–22, 2016, Kyiv, Ukraine), Book of abstracts, Kyiv (2016), p. 43.
I. Yu. Raievska, M. Yu. Raievska, Ya. P. Sysak, Local nearrings of order 243, Proceedings of the 4th Conference of Mathematical Society of Moldova CMSM4'2017 (June 28–July 2, 2017), Chisinau, Republic of Moldova (2017), p. 149–152.
I. Raievska, M. Raievska, Ya. Sysak, DatabaseEndom128: (v0.2) [Data set] (2022); https://doi.org/10.5281/ zenodo.6538441.
I. Raievska, M. Raievska, Ya. Sysak, DatabaseEndom625: (v0.2) [Data set] (2023); https://zenodo.org/record/ 7613145#.ZChqJXZBy39.
I. Raievska, M. Raievska, Ya. Sysak, DatabaseEndom32: (v1.0.2) [Data set] (2024); https://zenodo.org/records/ 10820301.
I. Raievska, M. Raievska, Ya. Sysak, DatabaseEndom64: (v1.0.1) [Data set] (2024); https://zenodo.org/records/ 10820320.
I. Raievska, M. Raievska, Ya. Sysak, DatabaseEndom243: (v1.0.1) [Data set] (2024); https://zenodo.org/records/ 10820318.
I. Yu. Raievska, M. Yu. Raievska, Ya. P. Sysak, Semidistributive nearrings with identity, Algebra and Discrete Math., 36, № 2 (2023). DOI: https://doi.org/10.12958/adm2207
I. Raievska, M. Raievska, Ya. Sysak, LocalNR, Package of local nearrings, Version 1.0.4 (2024) (GAP package); https://gap-packages.github.io/LocalNR/. DOI: https://doi.org/10.18523/2617-7080i2018p38-48
M. Raievska, On semidistributive local nearrings, Наук. вісн. Ужгород. університету. Математика і інформатика, 43, № 2, 67–71 (2023).
L. Redei, Das “schiefe Produkt” in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehören, Comment. Math. Helv., 20, 225–264 (1947). DOI: https://doi.org/10.1007/BF02568131
W. Rump, Set-theoretic solutions to the Yang–Baxter equation, skew-braces, and related near-rings, J. Algebra and Appl., 18, № 8, Article ID 1950145 (2019). DOI: https://doi.org/10.1142/S0219498819501457
Ya. P. Sysak, Products of groups and local nearrings, Note Mat., 28, № 2, 179–213 (2008).
Ya. P. Sysak, S. Di Termini, Local nearrings with generalized quaternion multiplicative group, Ric. Mat., 56, 61–72 (2007). DOI: https://doi.org/10.1007/s11587-007-0005-6
O. Taussky-Todd, Rings with non-commutative addition, Bull. Calcutta Math. Soc., 28, 245–246 (1936).
Wähling H, Theorie der Fastkörper, Thales-Verlag, Essen (1987).
H. Zassenhaus, Über endliche Fastkörper, Abh. Math. Sem., Univ. Hamburg, 11, 187–220 (1935/36). DOI: https://doi.org/10.1007/BF02940723
І. Ю. Раєвська, Локальні майже-кільця на неметациклічній $p$-групі Міллера–Морено, Вісн. Київ. ун-ту. Математика, механіка, 25, 43–45 (2011).
І. Ю. Раєвська, М. Ю. Раєвська, Локальні майже-кільця з обмеженнями на мультиплікативні групи та підгрупи необоротних елементів, Наук. часопис НПУ імені М. П. Драгоманова. Фіз.-мат. науки, № 14, 134–145 (2013).
І. Ю. Раєвська, М. Ю. Раєвська, Скiнченнi локальнi майже-кiльця, Могилян. мат. журн., 1, 38–48 (2018).
І. Ю. Раєвська, М. Ю. Раєвська, Локальні майже-кільця з мультиплікативною групою Шмідта, Укр. мат. журн., 71, № 10, 1435–1440 (2019); English translation: Ukr. Math. J., 71, № 10, 1643–1649 (2020).
І. Ю. Раєвська, М. Ю. Раєвська, Локальні майже-кільця на елементарних абелевих групах порядку $p^3$, Наук. вісн. Ужгород. ун-у. Математика і інформатика, 38, № 1, 85–93 (2021). DOI: https://doi.org/10.24144/2616-7700.2021.38(1).85-93
І. Ю. Раєвська, М. Ю. Раєвська, Групи порядку $p^4$ як адитивні групи локальних майже-кілець, Укр. мат. журн., 76, № 6, 890–-906 (2024). DOI: https://doi.org/10.3842/umzh.v76i5.8053
І. Ю. Раєвська, М. Ю. Раєвська, Я. П. Сисак, Локальні майже-кільця на неметациклічних групах Міллера–Морено, Вісн. Київ. ун-ту. Фіз.-мат. науки, 3, 39–46 (2012).
М. Ю. Раєвська, Локальні майже-кільця з мультиплікативною групою Міллера–Морено, Вісн. Київ. ун-ту. Математика, механіка, 25, 45–48 (2011).
М. Ю. Раєвська, Я. П. Сисак, Про локальні майже-кільця з мультиплікативною групою Міллера–Морено, Укр. мат. журн., 64, № 6, 811–818 (2012); English translation: Ukr. Math. J., 64, № 6, 930–937 (2012).
В. М. Усенко, Напівгрупи та майжекільця перетворень, автореф. дис... д-ра фіз.-мат. наук, Київ (1999).
О. Ю. Шмидт, Группы, все подгруппы которых специальные, Мат. сб., 31, 366–372 (1924).
Copyright (c) 2024 Maryna Raievska
This work is licensed under a Creative Commons Attribution 4.0 International License.
