2F-modules, abelian sets of roots and 2-ranks

We complete the classification of the irreducible 2F-modules for quasi-simple groups by dealing with the two open cases for sporadic groups from [R.M. Guralnick, G. Malle, Classification of 2F-modules, I, J. Algebra 257 (2002) 348–372]. The techniques developed involve abelian sets of roots within root systems, and enable us to determine the 2-ranks of the Monster, the Baby Monster and the double cover of the latter.