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''(in group theory)''

Let  $  p > 0 $
be a prime number. The  $  p $-
rank of a [[Finite group|finite group]]  $  G $
is the largest integer  $  n $
such that  $  G $
has an elementary Abelian subgroup of order  $  p  ^ {n} $(
cf. [[Abelian group|Abelian group]]). A  $  p $-
group is elementary Abelian if it is a direct product of cyclic groups of order  $  p $(
cf. [[Cyclic group|Cyclic group]]). A finite group  $  G $
has  $  p $-
rank  $  1 $
if and only if either the Sylow  $  p $-
subgroup (cf. [[Sylow subgroup|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|Normal subgroup]]). The sectional  $  p $-
rank of  $  G $
is the maximum of the  $  p $-
ranks of the Abelian sections  $  B/A $
for subgroups  $  A 	riangle\left B \right .$
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|Simple finite group]]). Some details can be found in [[#References|[a2]]] and [[#References|[a3]]]. In particular, see [[#References|[a3]]], Sect. 1.5. In [[#References|[a2]]], the word  "p-depth of a groupdepth"  is used and  "rank"  is reserved for a different concept. In the [[Cohomology of groups|cohomology of groups]], a celebrated theorem of D. Quillen [[#References|[a4]]] states that the  $  p $-
rank of  $  G $
is the same as the Krull dimension (cf. [[Dimension|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 [[#References|[a1]]]).

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