View source for P-rank

''(in group theory)''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p1100203.png" /> be a prime number. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p1100204.png" />-rank of a [[Finite group|finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p1100205.png" /> is the largest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p1100206.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p1100207.png" /> has an elementary Abelian subgroup of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p1100208.png" /> (cf. [[Abelian group|Abelian group]]). A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p1100209.png" />-group is elementary Abelian if it is a direct product of cyclic groups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002011.png" /> (cf. [[Cyclic group|Cyclic group]]). A finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002012.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002013.png" />-rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002014.png" /> if and only if either the Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002015.png" />-subgroup (cf. [[Sylow subgroup|Sylow subgroup]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002016.png" /> is cyclic or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002017.png" /> and the Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002018.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002019.png" /> is generalized quarternion. There are several variations on the definition. For example, the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002021.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002022.png" /> is the maximum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002023.png" />-ranks of the Abelian normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002024.png" /> (cf. [[Normal subgroup|Normal subgroup]]). The sectional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002026.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002027.png" /> is the maximum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002028.png" />-ranks of the Abelian sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002029.png" /> for subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002031.png" />.

The notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002032.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002033.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002034.png" /> is the same as the Krull dimension (cf. [[Dimension|Dimension]]) of the modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002035.png" /> cohomology ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002036.png" />. The connection can be described as follows. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002037.png" /> is a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002038.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002039.png" /> be an elementary Abelian subgroup of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002040.png" />. By direct calculation it can be shown that the cohomology ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002041.png" /> modulo its radical is a polynomial ring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002042.png" /> variables. Hence its maximal ideal spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002043.png" /> is an affine space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002044.png" />. Quillen's theorem says that the restriction mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002045.png" /> induces a finite-to-one mapping of varieties

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002046.png" /></td> </tr></table>

and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002047.png" /> is the union of the images for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002048.png" />. Therefore, the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002049.png" />, which is the Krull dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002050.png" />, is the maximum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002051.png" />-ranks of the subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110020/p11002052.png" />. 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>