Fixed Point Spaces and Abelian Subgroups

In this chapter, we will find upper bounds for the dimension of the fixed point spaces of elements in metrically primitive irreducible solvable subgroups of \(\varDelta (V, \kappa )\) . We will do this by finding such upper bounds for elements of prime order in \(\operatorname {GL}_n(q)\) that normalize an absolutely irreducible extraspecial r-group of exponent \(r \gcd (r,2)\) . As a consequence, we prove the following result: if \(G \leq \operatorname {GL}_n(q)\) is irreducible solvable, then every abelian subgroup of the affine group \(\mathbb {F}_q^n \rtimes G\) has order \(\leq q^n\) . The results of this chapter will be applied in the next chapter, where we will complete the classification of maximal irreducible solvable subgroups.