Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 74, No. 3, pp. 869-880, 2024

On zero-symmetric nearrings with identity whose additive groups are simple

Wen-Fong Ke, Johannes H. Meyer, Günter F. Pilz, Gerhard Wendt

Received March 2, 2024. Published online July 24, 2024.

Abstract: We investigate conditions on an infinite simple group in order to construct a zero-symmetric nearring with identity on it. Using the Higman-Neumann-Neumann extensions and Clay's characterization, we obtain zero-symmetric nearrings with identity with the additive groups infinite simple groups. We also show that no zero-symmetric nearring with identity can have the symmetric group ${\rm Sym}(\mathbb{N})$ as its additive group.

Keywords: infinite simple group; HNN extension; nearring with identity

Classification MSC: 20E06, 16Y30, 20E32, 20B30

DOI: 10.21136/CMJ.2024.0086-24

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] G. Baumslag: Topics in Combinatorial Group Theory. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (1993). DOI 10.1007/978-3-0348-8587-4 | MR 1243634 | Zbl 0797.20001
[2] J. R. Clay: The near-rings on groups of low order. Math. Z. 104 (1968), 364-371. DOI 10.1007/BF01110428 | MR 0224659 | Zbl 0153.35704
[3] J. R. Clay, J. J. Malone, Jr.: The near-rings with identities on certain finite groups. Math. Scand. 19 (1966), 146-150. DOI 10.7146/math.scand.a-10803 | MR 0207774 | Zbl 0149.02701
[4] L. E. Dickson: Theory of linear groups in an arbitrary field. Trans. Am. Math. Soc. 2 (1901), 363-394. DOI 10.1090/S0002-9947-1901-1500573-3 | MR 1500573 | JFM 32.0131.03
[5] J. D. Dixon, P. M. Neumann, S. Thomas: Subgroups of small index in infinite symmetric groups. Bull. Lond. Math. Soc. 18 (1986), 580-586. DOI 10.1112/blms/18.6.580 | MR 0859950 | Zbl 0607.20003
[6] A. G. Hamilton: Numbers, Sets and Axioms: The Apparatus of Mathematics. Cambridge University Press, Cambridge (1982). DOI 10.1017/cbo9781139171618 | MR 0691672 | Zbl 0497.04001
[7] G. Higman, B. H. Neumann, H. Neumann: Embedding theorems for groups. J. Lond. Math. Soc. 24 (1949), 247-254. DOI 10.1112/jlms/s1-24.4.247 | MR 0032641 | Zbl 0034.30101
[8] K. Kaarli: On ideal transitivity in near-rings. Contributions to General Algebra 8. Höder-Pichler-Tempsky, Vienna (1992), 81-89. MR 1281831 | Zbl 0790.16036
[9] R. C. Lyndon, P. E. Shupp: Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 89. Springer, Berlin (1977). DOI 10.1007/978-3-642-61896-3 | MR 0577064 | Zbl 0368.20023
[10] A. Y. Ol'shanskii: An infinite simple Noetherian group without torsion. Math. USSR, Izv. 15 (1980), 531-588; Translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43 (1979), 1328-1393. DOI 10.1070/IM1980v015n03ABEH001268 | MR 0567039 | Zbl 0453.20024
[11] A. Y. Ol'shanskii: Groups of bounded period with subgroups of prime order. Algebra Logic 21 (1983), 369-418; Translation from Algebra Logika 21 (1982), 553-618. DOI 10.1007/BF02027230 | MR 0721048 | Zbl 0524.20024
[12] G. Pilz: Near-Rings: The Theory and Its Applications. North-Holland Mathematics Studies 23. North-Holland, Amsterdam (1977). DOI 10.1016/s0304-0208(08)x7135-x | MR 0469981 | Zbl 0349.16015
[13] J. J. Rotman: An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148. Springer, Berlin (1995). DOI 10.1007/978-1-4612-4176-8 | MR 1307623 | Zbl 0810.20001
[14] J. Schreier, S. Ulam: Über die Permutationsgruppe der natürlichen Zahlenfolge. Stud. Math. 4 (1933), 134-141. (In German.) DOI 10.4064/sm-4-1-134-141 | Zbl 0008.20003
[15] J. Schreier, S. Ulam: Über die Automorphismen der Permutationsgruppe der natürlichen Zahlenfolge. Fundam. Math. 28 (1937), 258-260. (In German.) DOI 10.4064/fm-28-1-258-260 | Zbl 0016.20301
[16] E. A. Scott: A tour around finitely presented infinite simple groups. Algorithms and Classification in Combinatorial Group Theory. Springer, New York (1992), 83-119. DOI 10.1007/978-1-4613-9730-4_4 | MR 1230630 | Zbl 0753.20008
[17] W. R. Scott: Group Theory. Dover, New York 1987. MR 0896269 | Zbl 0641.20001

Affiliations: Wen-Fong Ke (correspponding author), Department of Mathematics, National Cheng Kung University, 1st, Dasyue Rd, East District, Tainan 701, Taiwan, e-mail: wfke@mail.ncku.edu.tw; Johannes H. Meyer, Department of Mathematics and Applied Mathematics, University of the Free State, PO Box 339, Bloemfontein, 9300, South Africa, e-mail: meyerjh@ufs.ac.za; Günter F. Pilz, Gerhard Wendt, Department of Algebra, Johannes Kepler Universität Linz, Altenberger Strasse 69, 4040 Linz, Austria, e-mail: guenter.pilz@jku.at, gerhard.wendt@gmx.at

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