Article
| Title: | On zero-symmetric nearrings with identity whose additive groups are simple (English) |
| Author: | Ke, Wen-Fong |
| Author: | Meyer, Johannes H. |
| Author: | Pilz, Günter F. |
| Author: | Wendt, Gerhard |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 3 |
| Year: | 2024 |
| Pages: | 869-880 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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. (English) |
| Keyword: | infinite simple group |
| Keyword: | HNN extension |
| Keyword: | nearring with identity |
| MSC: | 16Y30 |
| MSC: | 20B30 |
| MSC: | 20E06 |
| MSC: | 20E32 |
| DOI: | 10.21136/CMJ.2024.0086-24 |
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| Date available: | 2024-10-03T12:38:50Z |
| Last updated: | 2024-10-04 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152585 |
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| Reference: | [1] Baumslag, G.: Topics in Combinatorial Group Theory.Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (1993). Zbl 0797.20001, MR 1243634, 10.1007/978-3-0348-8587-4 |
| Reference: | [2] Clay, J. R.: The near-rings on groups of low order.Math. Z. 104 (1968), 364-371. Zbl 0153.35704, MR 0224659, 10.1007/BF01110428 |
| Reference: | [3] J. R. Clay, J. J. Malone, Jr.: The near-rings with identities on certain finite groups.Math. Scand. 19 (1966), 146-150. Zbl 0149.02701, MR 0207774, 10.7146/math.scand.a-10803 |
| Reference: | [4] Dickson, L. E.: Theory of linear groups in an arbitrary field.Trans. Am. Math. Soc. 2 (1901), 363-394 \99999JFM99999 32.0131.03. MR 1500573, 10.1090/S0002-9947-1901-1500573-3 |
| Reference: | [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. Zbl 0607.20003, MR 0859950, 10.1112/blms/18.6.580 |
| Reference: | [6] Hamilton, A. G.: Numbers, Sets and Axioms: The Apparatus of Mathematics.Cambridge University Press, Cambridge (1982). Zbl 0497.04001, MR 0691672, 10.1017/cbo9781139171618 |
| Reference: | [7] Higman, G., Neumann, B. H., Neumann, H.: Embedding theorems for groups.J. Lond. Math. Soc. 24 (1949), 247-254. Zbl 0034.30101, MR 0032641, 10.1112/jlms/s1-24.4.247 |
| Reference: | [8] Kaarli, K.: On ideal transitivity in near-rings.Contributions to General Algebra 8 Höder-Pichler-Tempsky, Vienna (1992), 81-89. Zbl 0790.16036, MR 1281831 |
| Reference: | [9] Lyndon, R. C., Shupp, P. E.: Combinatorial Group Theory.Ergebnisse der Mathematik und ihrer Grenzgebiete 89. Springer, Berlin (1977). Zbl 0368.20023, MR 0577064, 10.1007/978-3-642-61896-3 |
| Reference: | [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. Zbl 0453.20024, MR 0567039, 10.1070/IM1980v015n03ABEH001268 |
| Reference: | [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. Zbl 0524.20024, MR 0721048, 10.1007/BF02027230 |
| Reference: | [12] Pilz, G.: Near-Rings: The Theory and Its Applications.North-Holland Mathematics Studies 23. North-Holland, Amsterdam (1977). Zbl 0349.16015, MR 0469981, 10.1016/s0304-0208(08)x7135-x |
| Reference: | [13] Rotman, J. J.: An Introduction to the Theory of Groups.Graduate Texts in Mathematics 148. Springer, Berlin (1995). Zbl 0810.20001, MR 1307623, 10.1007/978-1-4612-4176-8 |
| Reference: | [14] Schreier, J., Ulam, S.: Über die Permutationsgruppe der natürlichen Zahlenfolge.Stud. Math. 4 (1933), 134-141 German. Zbl 0008.20003, 10.4064/sm-4-1-134-141 |
| Reference: | [15] Schreier, J., Ulam, S.: Über die Automorphismen der Permutationsgruppe der natürlichen Zahlenfolge.Fundam. Math. 28 (1937), 258-260 German. Zbl 0016.20301, 10.4064/fm-28-1-258-260 |
| Reference: | [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. Zbl 0753.20008, MR 1230630, 10.1007/978-1-4613-9730-4_4 |
| Reference: | [17] Scott, W. R.: Group Theory.Dover, New York 1987 \99999MR99999 0896269 . Zbl 0641.20001, MR 0896269 |
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