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Functions on groups.

J. Ecker showed that a Frobenius group $ G$ is $ 1$-affine complete (i.e. every unary congruence compatible function is a polynomial function) if and only if its Frobenius kernel is $ 1$-affine complete and $ G$ is a generalized dihedral group. There are no $ k$-affine complete Frobenius groups for $ k>1$ [#!Ec:TF1A!#].

Descriptions of the unary polynomial functions on Frobenius complements and formulas for their numbers are given in [#!Ma:TPFO!#]. This involves a study of functions on SL$ (2,5)$. More generally in [#!AM:PFAE!#], E. Aichinger and P. Mayr described the polynomial functions and endomorphism near-rings on subgroups of the general linear groups that contain the special linear groups. In his Ph.D.-thesis, P. Mayr gave an account of polynomial functions on all non-solvable classical linear groups (unitary, symplectic, orthogonal).

The results on polynomial functions found widespread interest at international conferences on group theory and universal algebra. Also people from algebraic geometry and topology could be won to collaborate on functions on infinite linear groups [#!LM:ATPO!#].

Peter Mayr 2005-11-25