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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d1100901.png" /> be a [[Commutative ring|commutative ring]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d1100902.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d1100903.png" />-algebra, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d1100904.png" /> be a [[Group|group]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d1100905.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d1100907.png" />-algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d1100908.png" /> acts as a group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d1100909.png" />-algebra automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009010.png" />. Expressed otherwise, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009011.png" /> acts on each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009012.png" /> to give <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009013.png" /> such that this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009014.png" />-action makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009015.png" /> into a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009016.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009019.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009020.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009021.png" />. It is customary to write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009022.png" /> for the subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009023.png" /> consisting of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009024.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009026.png" />. The [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009027.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009028.png" />-algebra. Indeed, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009030.png" />, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009031.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009032.png" /> becomes a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009033.png" />-algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009034.png" />. Further examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009035.png" />-algebras can be found in [[#References|[a5]]].
+
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$#C+1 = 122 : ~/encyclopedia/old_files/data/D110/D.1100090 Defect group of a block
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009036.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009037.png" />-algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009039.png" /> is a [[Finite group|finite group]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009040.png" /> be subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009041.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009042.png" /> be a left transversal for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009044.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009045.png" /> picks out one element in each (e.g., left) coset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009047.png" />; thus, it is a transversal of the system of subsets consisting of the (left) cosets; cf. also [[Combinatorial analysis|Combinatorial analysis]]). Then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009049.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009050.png" />-linear and is independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009051.png" />. It is customary to refer to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009052.png" /> as the relative trace mapping and to denote the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009053.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009054.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009055.png" /> be a primitive idempotent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009056.png" />. A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009058.png" /> is called a defect group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009059.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009060.png" /> is a minimal element in the set of subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009062.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009063.png" />. Such a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009064.png" /> exists because the set of all subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009065.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009066.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009067.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009068.png" />, and hence is non-empty. In the special case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009069.png" /> it is clear that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009070.png" /> is a block idempotent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009071.png" /> (cf. [[Block|Block]]). For this reason the defect groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009072.png" /> are also called the defect groups of the block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009073.png" />.
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Again let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009074.png" /> be a finite group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009075.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009076.png" />-algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009077.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009078.png" /> be a primitive idempotent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009079.png" />. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009080.png" /> a defect group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009081.png" /> and assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009082.png" /> is semi-perfect (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009083.png" /> is a finitely generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009084.png" />-module and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009085.png" /> is a complete Noetherian semi-local ring, cf. also [[Noetherian ring|Noetherian ring]]; [[Local ring|Local ring]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009086.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009087.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009088.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009089.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009090.png" />. Moreover, a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009091.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009092.png" /> is a defect group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009093.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009094.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009095.png" />-conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009096.png" />. In the most important case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009097.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009098.png" /> a prime number), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d11009099.png" /> turns out to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090100.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090101.png" /> (see [[#References|[a5]]]).
+
Let  $  R $
 +
be a [[Commutative ring|commutative ring]], let $  A $
 +
be an  $  R $-
 +
algebra, and let $  G $
 +
be a [[Group|group]]. Then  $  A $
 +
is said to be  $  G $-
 +
algebra if  $  G $
 +
acts as a group of $  R $-
 +
algebra automorphisms of  $  A $.  
 +
Expressed otherwise, each  $  g \in G $
 +
acts on each  $  a \in A $
 +
to give  $  ^ {g} a \in A $
 +
such that this  $  G $-
 +
action makes  $  A $
 +
into a left  $  RG $-
 +
module and  $  ^ {g} ( ab ) = {}  ^ {g} a {}  ^ {g} b $
 +
for all  $  a,b \in A $,  
 +
$  g \in G $.  
 +
Assume that  $  H $
 +
is a subgroup of  $  G $.  
 +
It is customary to write  $  A  ^ {H} $
 +
for the subalgebra of  $  A $
 +
consisting of all  $  a \in A $
 +
for which  $  ^ {h} a = a $
 +
for all  $  h \in H $.  
 +
The [[Group algebra|group algebra]] $  RG $
 +
is a $  G $-
 +
algebra. Indeed, for any  $  x \in RG $
 +
and  $  g \in G $,
 +
define  $  ^ {g} x = gxg ^ {- 1 } $.  
 +
Then  $  RG $
 +
becomes a $  G $-
 +
algebra over  $  R $.  
 +
Further examples of $  G $-
 +
algebras can be found in [[#References|[a5]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090102.png" /> be a complete Noetherian semi-local ring such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090103.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090104.png" /> a prime number) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090105.png" /> be a [[Block|block]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090106.png" />. Then a defect group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090107.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090108.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090109.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090110.png" /> is a defect group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090111.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090112.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090113.png" />. The integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090114.png" /> is called the defect of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090115.png" />. The study of defect groups of blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090116.png" /> is especially important in the following cases:
+
Let $  A $
 +
be a $  G $-
 +
algebra over  $  R $,
 +
where  $  G $
 +
is a [[Finite group|finite group]], let  $  H \subseteq K $
 +
be subgroups of  $  G $,
 +
and let  $  T $
 +
be a left transversal for  $  H $
 +
in  $  K $(
 +
i.e.,  $  T $
 +
picks out one element in each (e.g., left) coset of  $  H $
 +
in  $  K $;
 +
thus, it is a transversal of the system of subsets consisting of the (left) cosets; cf. also [[Combinatorial analysis|Combinatorial analysis]]). Then the mapping  $  { { \mathop{\rm Tr} } _ {H}  ^ {K} } : {A  ^ {H} } \rightarrow {A  ^ {K} } $,
 +
a \mapsto \sum _ {t \in T }  {}  ^ {t} a $,
 +
is  $  R $-
 +
linear and is independent of the choice of $  T $.  
 +
It is customary to refer to  $  { \mathop{\rm Tr} } _ {H}  ^ {K} $
 +
as the relative trace mapping and to denote the image of  $  { \mathop{\rm Tr} } _ {H}  ^ {K} $
 +
by  $  A _ {H}  ^ {K} $.  
 +
Let  $  e $
 +
be a primitive idempotent of  $  A  ^ {G} $.  
 +
A subgroup $  D $
 +
of $  G $
 +
is called a defect group of $  e $
 +
if  $  D $
 +
is a minimal element in the set of subgroups  $  H $
 +
of  $  G $
 +
such that  $  e \in A _ {H}  ^ {G} $.  
 +
Such a subgroup  $  D $
 +
exists because the set of all subgroups  $  H $
 +
of  $  G $
 +
with  $  e \in A _ {H}  ^ {G} $
 +
contains  $  G $,  
 +
and hence is non-empty. In the special case where  $  A = RG $
 +
it is clear that  $  e $
 +
is a block idempotent of  $  RG $(
 +
cf. [[Block|Block]]). For this reason the defect groups of $  e $
 +
are also called the defect groups of the block  $  B = RGe $.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090117.png" /> is a complete discrete valuation ring of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090118.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090119.png" /> of prime characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090120.png" />;
+
Again let  $  G $
 +
be a finite group, let  $  A $
 +
be a  $  G $-
 +
algebra over  $  R $,
 +
and let  $  e $
 +
be a primitive idempotent of  $  A  ^ {G} $.  
 +
Denote by  $  D $
 +
a defect group of  $  e $
 +
and assume that  $  A  ^ {G} $
 +
is semi-perfect (e.g.,  $  A $
 +
is a finitely generated  $  R $-
 +
module and  $  R $
 +
is a complete Noetherian semi-local ring, cf. also [[Noetherian ring|Noetherian ring]]; [[Local ring|Local ring]]). If  $  H $
 +
is a subgroup of $  G $
 +
such that  $  e \in A _ {H}  ^ {G} $,
 +
then  $  D \subseteq gHg ^ {- 1 } $
 +
for some  $  g \in G $.  
 +
Moreover, a subgroup  $  K $
 +
of  $  G $
 +
is a defect group of  $  e $
 +
if and only if  $  K $
 +
is  $  G $-
 +
conjugate to  $  D $.  
 +
In the most important case where  $  { \mathop{\rm char} } ( R/J ( R ) ) = p $(
 +
$  p $
 +
a prime number),  $  D $
 +
turns out to be a  $  p $-
 +
subgroup of $  G $(
 +
see [[#References|[a5]]]).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090121.png" /> is a field of prime characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090122.png" /> (see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]).
+
Let  $  R $
 +
be a complete Noetherian semi-local ring such that  $  { \mathop{\rm char} } ( R/J ( R ) ) = p $(
 +
$  p $
 +
a prime number) and let  $  B $
 +
be a [[Block|block]] of  $  RG $.
 +
Then a defect group of  $  B $
 +
is a  $  p $-
 +
subgroup of  $  G $.
 +
Thus, if  $  D $
 +
is a defect group of  $  B $,  
 +
then  $  | D | = p  ^ {d} $
 +
for some integer  $  d \geq  0 $.
 +
The integer  $  d $
 +
is called the defect of  $  B $.  
 +
The study of defect groups of blocks of  $  RG $
 +
is especially important in the following cases:
  
The ring-theoretic structure of blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110090/d110090123.png" /> is closely related to the structure of their defect groups. The most extensive coverage of this topic can be found in [[#References|[a5]]]. For various applications to the modular representation theory of groups, see also [[#References|[a5]]]. There is no doubt that the current theory of defect groups of blocks will be significantly strengthened in the future, and that the field contains untapped reserves rich enough to ensure major strikes. For various open problems and the current (1996) state of the subject, see [[#References|[a5]]].
+
$  R $
 +
is a complete discrete valuation ring of characteristic  $  0 $
 +
with  $  R/J ( R ) $
 +
of prime characteristic  $  p $;
 +
 
 +
$  R $
 +
is a field of prime characteristic  $  p $(
 +
see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]]).
 +
 
 +
The ring-theoretic structure of blocks of $  RG $
 +
is closely related to the structure of their defect groups. The most extensive coverage of this topic can be found in [[#References|[a5]]]. For various applications to the modular representation theory of groups, see also [[#References|[a5]]]. There is no doubt that the current theory of defect groups of blocks will be significantly strengthened in the future, and that the field contains untapped reserves rich enough to ensure major strikes. For various open problems and the current (1996) state of the subject, see [[#References|[a5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''1''' , North-Holland  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''2''' , North-Holland  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''3''' , North-Holland  (1994)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''4''' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''5''' , North-Holland  (1996)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''1''' , North-Holland  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''2''' , North-Holland  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''3''' , North-Holland  (1994)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''4''' , North-Holland  (1995)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Karpilovsky,  "Group representations" , '''5''' , North-Holland  (1996)</TD></TR></table>

Latest revision as of 17:32, 5 June 2020


Let $ R $ be a commutative ring, let $ A $ be an $ R $- algebra, and let $ G $ be a group. Then $ A $ is said to be $ G $- algebra if $ G $ acts as a group of $ R $- algebra automorphisms of $ A $. Expressed otherwise, each $ g \in G $ acts on each $ a \in A $ to give $ ^ {g} a \in A $ such that this $ G $- action makes $ A $ into a left $ RG $- module and $ ^ {g} ( ab ) = {} ^ {g} a {} ^ {g} b $ for all $ a,b \in A $, $ g \in G $. Assume that $ H $ is a subgroup of $ G $. It is customary to write $ A ^ {H} $ for the subalgebra of $ A $ consisting of all $ a \in A $ for which $ ^ {h} a = a $ for all $ h \in H $. The group algebra $ RG $ is a $ G $- algebra. Indeed, for any $ x \in RG $ and $ g \in G $, define $ ^ {g} x = gxg ^ {- 1 } $. Then $ RG $ becomes a $ G $- algebra over $ R $. Further examples of $ G $- algebras can be found in [a5].

Let $ A $ be a $ G $- algebra over $ R $, where $ G $ is a finite group, let $ H \subseteq K $ be subgroups of $ G $, and let $ T $ be a left transversal for $ H $ in $ K $( i.e., $ T $ picks out one element in each (e.g., left) coset of $ H $ in $ K $; thus, it is a transversal of the system of subsets consisting of the (left) cosets; cf. also Combinatorial analysis). Then the mapping $ { { \mathop{\rm Tr} } _ {H} ^ {K} } : {A ^ {H} } \rightarrow {A ^ {K} } $, $ a \mapsto \sum _ {t \in T } {} ^ {t} a $, is $ R $- linear and is independent of the choice of $ T $. It is customary to refer to $ { \mathop{\rm Tr} } _ {H} ^ {K} $ as the relative trace mapping and to denote the image of $ { \mathop{\rm Tr} } _ {H} ^ {K} $ by $ A _ {H} ^ {K} $. Let $ e $ be a primitive idempotent of $ A ^ {G} $. A subgroup $ D $ of $ G $ is called a defect group of $ e $ if $ D $ is a minimal element in the set of subgroups $ H $ of $ G $ such that $ e \in A _ {H} ^ {G} $. Such a subgroup $ D $ exists because the set of all subgroups $ H $ of $ G $ with $ e \in A _ {H} ^ {G} $ contains $ G $, and hence is non-empty. In the special case where $ A = RG $ it is clear that $ e $ is a block idempotent of $ RG $( cf. Block). For this reason the defect groups of $ e $ are also called the defect groups of the block $ B = RGe $.

Again let $ G $ be a finite group, let $ A $ be a $ G $- algebra over $ R $, and let $ e $ be a primitive idempotent of $ A ^ {G} $. Denote by $ D $ a defect group of $ e $ and assume that $ A ^ {G} $ is semi-perfect (e.g., $ A $ is a finitely generated $ R $- module and $ R $ is a complete Noetherian semi-local ring, cf. also Noetherian ring; Local ring). If $ H $ is a subgroup of $ G $ such that $ e \in A _ {H} ^ {G} $, then $ D \subseteq gHg ^ {- 1 } $ for some $ g \in G $. Moreover, a subgroup $ K $ of $ G $ is a defect group of $ e $ if and only if $ K $ is $ G $- conjugate to $ D $. In the most important case where $ { \mathop{\rm char} } ( R/J ( R ) ) = p $( $ p $ a prime number), $ D $ turns out to be a $ p $- subgroup of $ G $( see [a5]).

Let $ R $ be a complete Noetherian semi-local ring such that $ { \mathop{\rm char} } ( R/J ( R ) ) = p $( $ p $ a prime number) and let $ B $ be a block of $ RG $. Then a defect group of $ B $ is a $ p $- subgroup of $ G $. Thus, if $ D $ is a defect group of $ B $, then $ | D | = p ^ {d} $ for some integer $ d \geq 0 $. The integer $ d $ is called the defect of $ B $. The study of defect groups of blocks of $ RG $ is especially important in the following cases:

$ R $ is a complete discrete valuation ring of characteristic $ 0 $ with $ R/J ( R ) $ of prime characteristic $ p $;

$ R $ is a field of prime characteristic $ p $( see [a1], [a2], [a3], [a4], [a5]).

The ring-theoretic structure of blocks of $ RG $ is closely related to the structure of their defect groups. The most extensive coverage of this topic can be found in [a5]. For various applications to the modular representation theory of groups, see also [a5]. There is no doubt that the current theory of defect groups of blocks will be significantly strengthened in the future, and that the field contains untapped reserves rich enough to ensure major strikes. For various open problems and the current (1996) state of the subject, see [a5].

References

[a1] G. Karpilovsky, "Group representations" , 1 , North-Holland (1992)
[a2] G. Karpilovsky, "Group representations" , 2 , North-Holland (1993)
[a3] G. Karpilovsky, "Group representations" , 3 , North-Holland (1994)
[a4] G. Karpilovsky, "Group representations" , 4 , North-Holland (1995)
[a5] G. Karpilovsky, "Group representations" , 5 , North-Holland (1996)
How to Cite This Entry:
Defect group of a block. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defect_group_of_a_block&oldid=46601
This article was adapted from an original article by G. Karpilovsky (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article