# 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
How to Cite This Entry:
P-rank. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-rank&oldid=51117
This article was adapted from an original article by J. Carlson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article