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Finite automorphism groups of laminated near-rings

  • K. D. Magill (a1), P. R. Misra (a2) and U. B. Tewari

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In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=fPg. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

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References

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1.Magill, K. D. Jr., Semigroups and near-rings of continuous functions, General Topology and its Relations to Modern Analysis and Algebra, III, Proc. Third Prague Top. Symp. 1971 (Academia, 1972), 283288.
2.Magill, K. D. Jr., Automorphism groups of laminated near-rings, Proc. Edinburgh Math. Soc. 23 (1980), 97102.
3.Magill, K. D. Jr., Misra, P. R. and Tewari, U. B., Automorphism groups of laminated near-rings determined by complex polynomials, Proc. Edinburgh Math. Soc. 26 (1983), 7384.

Finite automorphism groups of laminated near-rings

  • K. D. Magill (a1), P. R. Misra (a2) and U. B. Tewari

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