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

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/√6 | V | a / 6 if V is a faithful and completely reducible G -module and H has abelian Sylow 2-subgroups, where a=3ln 6/4ln 2 a = 3 ln 6 4 ln 2 . Furthermore, we establish a similar bound under the additional condition that 2 |V| 2 ∤ | V | .