P-rank

(in group theory)

Let $ p > 0 $ be a prime number. The $ p $- rank of a finite group $ G $ is the largest integer $ n $ such that $ G $ has an elementary Abelian subgroup of order $ p ^ {n} $( cf. Abelian group). A $ p $- group is elementary Abelian if it is a direct product of cyclic groups of order $ p $( cf. Cyclic group). A finite group $ G $ has $ p $- rank $ 1 $ if and only if either the Sylow $ p $- subgroup (cf. Sylow subgroup) of $ G $ is cyclic or $ p = 2 $ and the Sylow $ p $- subgroup of $ G $ is generalized quarternion. There are several variations on the definition. For example, the normal $ p $- rank of $ G $ is the maximum of the $ p $- ranks of the Abelian normal subgroups of $ G $( cf. Normal subgroup). The sectional $ p $- rank of $ G $ is the maximum of the $ p $- ranks of the Abelian sections $ B/A $ for subgroups $ A \lhd B$ of $ G $.

The notion of $ p $- 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 $ p $- rank of $ G $ is the same as the Krull dimension (cf. Dimension) of the modulo $ p $ cohomology ring of $ G $. The connection can be described as follows. Suppose $ k $ is a field of characteristic $ p $. Let $ E $ be an elementary Abelian subgroup of order $ p ^ {n} $. By direct calculation it can be shown that the cohomology ring of $ E $ modulo its radical is a polynomial ring in $ n $ variables. Hence its maximal ideal spectrum $ V _ {E} ( k ) $ is an affine space of dimension $ n $. Quillen's theorem says that the restriction mapping $ { { \mathop{\rm res} } _ {G,E } } : {H ^ {*} ( G, k ) } \rightarrow {H ^ {*} ( E, k ) } $ induces a finite-to-one mapping of varieties

$$ { { \mathop{\rm res} } _ {G,E } ^ {*} } : {V _ {E} ( k ) } \rightarrow {V _ {G} ( k ) } $$

and, moreover, $ V _ {G} ( k ) $ is the union of the images for all $ E $. Therefore, the dimension of $ V _ {G} ( k ) $, which is the Krull dimension of $ H ^ {*} ( G, k ) $, is the maximum of the $ p $- ranks of the subgroups $ E $. 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