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More on automorphism groups of laminated near-rings

  • D. K. Blevins (a1), K. D. Magill (a2), P. R. Misra (a3), J. C. Parnami (a4) and U. B. Tewari (a5)...

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We will assume throughout this paper that polynomials are nonconstant. 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 multiplication is defined by fg = f ο P ο g for all f,gp. The near-ring p is referred to as a laminated near-ring and P is referred to as the laminating element or laminator. In [1] the problem was posed of determining Aut p the automorphism group of p. It was shown that exactly three infinite groups occur as automorphism groups of the laminated near-rings p and for each of the three groups those polynomials P were characterized such that Aut p is isomorphic to that particular group. The infinite groups turn out to be GL(2), the full linear group of all 2×2 nonsingular real matrices and two of its subgroups.

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References

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1.Magill, K. D., Misra, P. R. and Tewari, U. B., Automorphism groups of laminated near-rings determined by complex polynomials, Proc. Edinburgh Math. Soc. 26 (1983), 7384.
2.Magill, K. D., Misra, P. R. and Tewari, U. B., Finite automorphism groups of laminated near-rings, Proc. Edinburgh Math. Soc. 26 (1983), 297306.
3.Walsh, J. L., The Location of Critical Points of Analytic and Harmonic Functions (Colloquium Pub. Vol. 34, Amer. Math. Soc., New York, 1950).

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