Bounding the order of solvable linear groups with abelian Sylow 2-subgroups

Abstract In this paper, we prove that if H is a subgroup of a finite solvable group G , then H has order at most $$|V|^{a}/\sqrt{6}$$ | V | a / 6 if V is a faithful and completely reducible G -module and H has abelian Sylow 2-subgroups, where $$a=\frac{3\ln 6}{4\ln 2}$$ a = 3 ln 6 4 ln 2 . Furthermore, we establish a similar bound under the additional condition that $$2\not \mid |V|$$ 2 ∤ | V | .