Polynomial functions and endomorphism near-rings on certain linear groups

We describe the unary polynomial functions on the non-solvable groups $G$ with $\SL(n,q) \le G \le \GL(n,q)$ and on their quotients $G/Y$ with $Y \le Z(G)$, 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 $\SL(n,q) \le G \le \GL(n,q)$, 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 $G' \cap Z(G) < N < G'$.