Polynomial Functions and Endomorphism Near-Rings on Certain Linear Groups

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) = E(G/Y) holds. For the case Y = {1}, this characterization yields that we have E(G) = I(G) if and only if G = 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) = A(G). Our results are based on a description of the polynomial functions on those non-abelian finite groups G that satisfy the following conditions: G′ = G″, G/Z(G) is centerless, and there is no normal subgroup N of G with G′ ∩ Z(G) < N < G′.