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The endomorphism near-rings of the symmetric groups of degree at least five

Published online by Cambridge University Press:  09 April 2009

Y. Fong  and
J. D. P. Meldrum
Y. Fong
Affiliation:
Department of Mathematics King's Buildings, Mayfield Road Edinburgh EH9 3JZ, U.K.
J. D. P. Meldrum
Affiliation:
Department of Mathematics King's Buildings, Mayfield Road Edinburgh EH9 3JZ, U.K.
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Abstract

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The near-ring distributively generated by the semigroup of all endomorphisms of Sn, the symmetric group of degree n, for n ≥ 5, is close to being the near-ring of all mappings from Sn to itself respecting the identity. In this paper, the structure of these near-rings is studied in detail. In particular, addition and multiplication rules for the elements given in canonical form are determined. A complete list of all right ideals, left ideals, right invariant and left invariant subgroups is given.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 1 , September 1980 , pp. 37 - 49
Copyright
Copyright © Australian Mathematical Society 1980

References

Betsch, G. (1973), ‘Some structure theorems on 2-primitive near-rings’, in Rings, modules and radicals. Colloquia Mathematica Societatis János Bolyai 6 edited by Kertész, A., pp. 73–102 (North Holland).Google Scholar
Fong, Y. (1979), The endomorphism near-rings of the symmetric groups (Ph.D. Thesis, University of Edinburgh).Google Scholar
Heatherly, H. E. (1972), ‘One-sided ideals in near-rings of transformations’, J.Austral. Math. Soc. 13, 171–179.CrossRefGoogle Scholar
Laxton, R. R. (1963), ‘Primitive distributively generated near-rings’, J. London Math. Soc. 38, 40–49.CrossRefGoogle Scholar
Lyons, C. G. (1973), ‘On decompositions of E(G)’, Rocky Mountain J. Math. 3, 575–582.CrossRefGoogle Scholar
Lyons, C. G. and Meldrum, J. D. P. (1980a), ‘Reduction theorems for endomorphism near-rings’ Monatshefte für Mathematik (to appear).CrossRefGoogle Scholar
Lyons, C. G. and Meldrum, J. D. P. (1980b), ‘N-series and tame near-rings’. Proc. Roy. Soc. Edinburgh Sect. A (to appear).CrossRefGoogle Scholar
Malone, J. J. (1973), ‘Generalized quaternion groups and distributively generated near-rings’, Proc. Edinburgh Math. Soc. 18, 235–238.CrossRefGoogle Scholar
Malone, J. J. (1977), ‘More on groups in which each element commutes with its endomorphic image’, Proc. Amer. Math. Soc. 65, 209–214.CrossRefGoogle Scholar
Malone, J. J. (1980), ‘A non-abelian 2-group whose endomorphisms generate a ring, and other examples of E-groups’, Proc. Edinburgh Math. Soc. (to appear).CrossRefGoogle Scholar
Malone, J. J. and Lyons, C. G. (1970), ‘Endomorphism near-rings’, Proc. Edinburgh Math. Soc. 17, 71–78.CrossRefGoogle Scholar
Malone, J. J. and Lyons, C. G. (1972), ‘Finite dihedral groups and d.g. near-rings I’, Compositio Math. 24, 305–312.Google Scholar
Malone, J. J. and Lyons, C. G. (1973), ‘Finite dihedral groups and d.g. near-rings II’, Compositio Math. 26, 249–259.Google Scholar
Malone, J. J. and McQuarrie, B. (1970), ‘Endomorphism rings of non-abelian groups’, Bull. Austral. Math. Soc. 3, 349–352.Google Scholar
Meldrum, J. D. P. (1978), ‘On the structure of morphism near-rings’, Proc. Roy. Soc. Edinburgh Sect A 81, 287–298.CrossRefGoogle Scholar
Meldrum, J. D. P. (1979), ‘The endomorphism near-rings of finite general linear groups’, Proc. Roy. Irish Acad. Sect. A 79, 87–96.Google Scholar
Meldrum, J. D. P. and Lyons, C. G. (1978), ‘Characterizing series for faithful d.g. near-rings’, Proc. Amer. Math. Soc. 72, 221–227.Google Scholar
Pilz, G. (1977), Near-rings (North-Holland, Amsterdam; American Elsevier, New York).Google Scholar
Scott, W. R. (1964), Group theory (Prentice-Hall, Englewood Cliffs, N.J.).Google Scholar