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''(in group theory)''
 
''(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" />.
+
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 \lhd B$
 +
of $  G $.
  
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
+
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
  
<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>
+
$$
 +
{ { \mathop{\rm res} } _ {G,E }  ^ {*} } : {V _ {E} ( k ) } \rightarrow {V _ {G} ( k ) }
 +
$$
  
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]]]).
+
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====
 
====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>
 
<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>

Latest revision as of 14:10, 31 December 2020


(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=13520
This article was adapted from an original article by J. Carlson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article