"Polynomial Functions on Classical Groups and Frobenius Groups"
, Universität Linz, 3-2004
Polynomial Functions on Classical Groups and Frobenius Groups
Sprache des Titels:
The intuitive concept of polynomial functions on groups (or on arbitrary algebras) is rather straightforward: a polynomial function is a function that can be expressed by a certain term.
We consider the following problems:
(1) How many unary polynomial functions are there on a given group?
(2) Are all automorphisms on a given group polynomial functions?
(3) Are all endomorphisms on a given group polynomial functions?
First we characterize the unary polynomial functions on the finite
groups whose quotient by the center has a non-abelian unique minimal
normal subgroup. This description generalizes a number of results
by A. Froehlich and others on non-solvable groups.
A crucial tool in our proof is interpolation of functions.
As a consequence of this characterization we are able to give complete
solutions of the problems (1), (2), and (3) for classical groups -
finite linear, unitary, symplectic, and orthogonal groups (with the
exception of certain groups acting on vector spaces of low dimension).
In the second part of this work, we start with E. Aichinger's description of unary polynomial functions on certain semidirect products that have a Frobenius group as quotient. Then we determine the number of polynomial functions on the finite solvable groups all of whose abelian subgroups are cyclic. For Frobenius complements, we give a full solution of problem (1), and partial solutions for problems (2), (3).