Polynomial Functions and Endomorphism Near-Rings on Certain Linear Groups Erhard Aichinger * and Peter Mayr * Institut fu ¨ r Algebra, Stochastik und wissensbasierte mathematische Systeme, Johannes Kepler Universita ¨t Linz, Linz, Austria ABSTRACT We describe the unary polynomial functions on the non-solvable groups G with SL(n, q)  G  GL(n, q) and on their quotients G=Y with Y  Z(G ), and we compute the size of the inner automorphism near-ring I(G=Y). We compare this near-ring to the endomorphism near-ring E(G=Y), and we obtain a full characterization of those G and Y for which I(G=Y) ¼ E(G=Y) holds. For the case Y ¼f1g, this characterization yields that we have E(G ) ¼ I(G ) if and only if G ¼ SL(n, q). We investigate the automorphism near-ring A(G ), and we show that for all non-solvable groups G with SL(n, q)  G  GL(n, q), we have I(G ) ¼ A(G ). Our results are based on a description of the polynomial functions on those non-abelian finite *Correspondence: Erhard Aichinger and Peter Mayr, Institut fu ¨ r Algebra, Sto- chastik und wissensbasierte mathematische Systeme, Johannes Kepler Universita ¨t Linz, 4040 Linz, Austria; E-mail: erhard@algebra.uni-linz.ac.at and peter.mayr@ algebra.uni-linz.ac.at. COMMUNICATIONS IN ALGEBRA Õ Vol. 31, No. 11, pp. 5627–5651, 2003 5627 DOI: 10.1081/AGB-120023978 0092-7872 (Print); 1532-4125 (Online) Copyright # 2003 by Marcel Dekker, Inc. www.dekker.com