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On zero-symmetric nearrings with identity whose additive groups are simple
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 869-880
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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.
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.
DOI : 10.21136/CMJ.2024.0086-24
Classification : 16Y30, 20B30, 20E06, 20E32
Keywords: infinite simple group; HNN extension; nearring with identity 
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Ke, Wen-Fong; Meyer, Johannes H.; Pilz, Günter F.; Wendt, Gerhard. On zero-symmetric nearrings with identity whose additive groups are simple. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 869-880. doi: 10.21136/CMJ.2024.0086-24

[1] Baumslag, G.: Topics in Combinatorial Group Theory. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (1993). | DOI | MR | JFM

[2] Clay, J. R.: The near-rings on groups of low order. Math. Z. 104 (1968), 364-371. | DOI | MR | JFM

[3] J. R. Clay, J. J. Malone, Jr.: The near-rings with identities on certain finite groups. Math. Scand. 19 (1966), 146-150. | DOI | MR | JFM

[4] Dickson, L. E.: Theory of linear groups in an arbitrary field. Trans. Am. Math. Soc. 2 (1901), 363-394 \99999JFM99999 32.0131.03. | DOI | MR

[5] Dixon, J. D., Neumann, P. M., Thomas, S.: Subgroups of small index in infinite symmetric groups. Bull. Lond. Math. Soc. 18 (1986), 580-586. | DOI | MR | JFM

[6] Hamilton, A. G.: Numbers, Sets and Axioms: The Apparatus of Mathematics. Cambridge University Press, Cambridge (1982). | DOI | MR | JFM

[7] Higman, G., Neumann, B. H., Neumann, H.: Embedding theorems for groups. J. Lond. Math. Soc. 24 (1949), 247-254. | DOI | MR | JFM

[8] Kaarli, K.: On ideal transitivity in near-rings. Contributions to General Algebra 8 Höder-Pichler-Tempsky, Vienna (1992), 81-89. | MR | JFM

[9] Lyndon, R. C., Shupp, P. E.: Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 89. Springer, Berlin (1977). | DOI | MR | JFM

[10] Ol'shanskii, A. Y.: 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 | MR | JFM

[11] Ol'shanskii, A. Y.: Groups of bounded period with subgroups of prime order. Algebra Logic 21 (1983), 369-418 Translation from Algebra Logika 21 1982 553-618. | DOI | MR | JFM

[12] Pilz, G.: Near-Rings: The Theory and Its Applications. North-Holland Mathematics Studies 23. North-Holland, Amsterdam (1977). | DOI | MR | JFM

[13] Rotman, J. J.: An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148. Springer, Berlin (1995). | DOI | MR | JFM

[14] Schreier, J., Ulam, S.: Über die Permutationsgruppe der natürlichen Zahlenfolge. Stud. Math. 4 (1933), 134-141 German. | DOI | JFM

[15] Schreier, J., Ulam, S.: Über die Automorphismen der Permutationsgruppe der natürlichen Zahlenfolge. Fundam. Math. 28 (1937), 258-260 German. | DOI | JFM

[16] Scott, E. A.: A tour around finitely presented infinite simple groups. Algorithms and Classification in Combinatorial Group Theory Springer, New York (1992), 83-119. | DOI | MR | JFM

[17] Scott, W. R.: Group Theory. Dover, New York 1987 \99999MR99999 0896269 . | MR | JFM

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