P-rank

(in group theory)

Let be a prime number. The -rank of a finite group is the largest integer such that has an elementary Abelian subgroup of order (cf. Abelian group). A -group is elementary Abelian if it is a direct product of cyclic groups of order (cf. Cyclic group). A finite group has -rank if and only if either the Sylow -subgroup (cf. Sylow subgroup) of is cyclic or and the Sylow -subgroup of is generalized quarternion. There are several variations on the definition. For example, the normal -rank of is the maximum of the -ranks of the Abelian normal subgroups of (cf. Normal subgroup). The sectional -rank of is the maximum of the -ranks of the Abelian sections for subgroups of .

The notion of -rank was used extensively to sort out cases in the classification of finite simple groups (cf. Simple finite group). Some details can be found in [a2] and [a3]. In particular, see [a3], Sect. 1.5. In [a2], the word "p-depth of a groupdepth" is used and "rank" is reserved for a different concept. In the cohomology of groups, a celebrated theorem of D. Quillen [a4] states that the -rank of is the same as the Krull dimension (cf. Dimension) of the modulo cohomology ring of . The connection can be described as follows. Suppose is a field of characteristic . Let be an elementary Abelian subgroup of order . By direct calculation it can be shown that the cohomology ring of modulo its radical is a polynomial ring in variables. Hence its maximal ideal spectrum is an affine space of dimension . Quillen's theorem says that the restriction mapping induces a finite-to-one mapping of varieties

and, moreover, is the union of the images for all . Therefore, the dimension of , which is the Krull dimension of , is the maximum of the -ranks of the subgroups . The theorem has found many applications in modular representation theory (see [a1]).

References

[a1]	D. J. Benson, "Representations and cohomology II: cohomology of groups and modules" , Cambridge Univ. Press (1991)
[a2]	D. Gorenstein, "Finite groups" , Harper and Row (1968)
[a3]	D. Gorenstein, "Finite groups" , Plenum (1982)
[a4]	D.G. Quillen, "The spectrum of an equivalent cohomology ring" Ann. of Math. , 94 (1971) pp. 549–602