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A group containing the given group as a [[Normal subgroup|normal subgroup]]. The quotient group is usually prescribed as well, that is, an extension of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e0369801.png" /> by a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e0369802.png" /> is a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e0369803.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e0369804.png" /> as a normal subgroup and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e0369805.png" />, i.e. it is an exact sequence
+
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<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/e/e036/e036980/e0369806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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 +
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In the literature other terminology is sometimes adopted, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e0369807.png" /> may be called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e0369808.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e0369809.png" /> (see [[#References|[2]]], for example), the epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698010.png" /> itself may be called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698011.png" /> (see [[#References|[1]]]), or the exact sequence (1) may be called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698012.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698013.png" />, or an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698014.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698015.png" />. An extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698016.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698017.png" /> always exists, although it is not uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698019.png" />. The need to describe all extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698020.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698021.png" /> up to a natural equivalence is motivated by the demands both of group theory itself and of its applications. Two extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698022.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698023.png" /> are called equivalent if there is a commutative diagram
+
A group containing the given group as a [[Normal subgroup|normal subgroup]]. The quotient group is usually prescribed as well, that is, an extension of a group  $  A $
 +
by a group  $  B $
 +
is a group $  G $
 +
containing  $  A $
 +
as a normal subgroup and such that  $  G/A \cong B $,
 +
i.e. it is an exact sequence
  
<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/e/e036/e036980/e03698024.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
e  \rightarrow  A  \rightarrow  G  \mathop \rightarrow \limits ^  \gamma    B  \rightarrow  e.
 +
$$
  
Any extension of the form (1) determines, via conjugation of the elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698025.png" />, a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698027.png" /> is the automorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698028.png" />,
+
In the literature other terminology is sometimes adopted, e.g.,  $  G $
 +
may be called an extension of $  B $
 +
by  $  A $(
 +
see [[#References|[2]]], for example), the epimorphism  $  \gamma :  G \rightarrow B $
 +
itself may be called an extension of  $  B $(
 +
see [[#References|[1]]]), or the exact sequence (1) may be called an extension of $  A $
 +
by  $  B $,  
 +
or an extension of  $  B $
 +
by  $  A $.  
 +
An extension of  $  A $
 +
by  $  B $
 +
always exists, although it is not uniquely determined by  $  A $
 +
and  $  B $.  
 +
The need to describe all extensions of  $  A $
 +
by  $  B $
 +
up to a natural equivalence is motivated by the demands both of group theory itself and of its applications. Two extensions of  $  A $
 +
by  $  B $
 +
are called equivalent if there is a commutative diagram
  
<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/e/e036/e036980/e03698029.png" /></td> </tr></table>
+
$$
 +
\begin{array}{c}
 +
e \rightarrow  \\
 +
{} \\
 +
e  \rightarrow 
 +
\end{array}
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698030.png" /> is contained in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698031.png" /> of inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698032.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698033.png" /> induces a homomorphism
+
\begin{array}{l}
 +
A  \rightarrow  \\
 +
\| \\
 +
A  \rightarrow 
 +
\end{array}
  
<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/e/e036/e036980/e03698034.png" /></td> </tr></table>
+
\begin{array}{l}
 +
G  \rightarrow  \\
 +
\downarrow \\
 +
G  ^  \prime  \rightarrow 
 +
\end{array}
  
The triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698035.png" /> is called the abstract kernel of the extension. Given an extension (1), one chooses for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698036.png" /> a representative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698037.png" /> in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698039.png" />. Then conjugation by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698040.png" /> determines an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698042.png" />,
+
\begin{array}{l}
 +
B  \rightarrow  \\
 +
\| \\
 +
B  \rightarrow 
 +
\end{array}
  
<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/e/e036/e036980/e03698043.png" /></td> </tr></table>
+
\begin{array}{l}
 +
e \\
 +
{} \\
 +
e
 +
\end{array}
  
The product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698045.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698046.png" /> up to a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698047.png" />:
+
$$
  
<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/e/e036/e036980/e03698048.png" /></td> </tr></table>
+
Any extension of the form (1) determines, via [[conjugation]] of the elements of the group  $  G $,
 +
a homomorphism  $  \alpha : G \rightarrow  \mathop{\rm Aut}  A $,
 +
where  $  \mathop{\rm Aut}  A $
 +
is the [[automorphism group]] of  $  A $,
  
It is easily checked that these functions must satisfy the conditions
+
$$
 +
\alpha(g) a = g a g^{-1} ,
 +
$$
 +
 
 +
such that  $  \alpha ( A) $
 +
is contained in the group  $  \mathop{\rm Inn}  A $
 +
of [[inner automorphism]]s of  $  A $.
 +
Hence  $  \alpha $
 +
induces a homomorphism
 +
 
 +
$$
 +
\beta :  B  \rightarrow \
 +
\mathop{\rm Aut}  A /  \mathop{\rm Inn}  A.
 +
$$
  
<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/e/e036/e036980/e03698049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
The triple  $  ( A, B, \beta ) $
 +
is called the abstract kernel of the extension. Given an extension (1), one chooses for every  $  b \in B $
 +
a representative  $  u ( b) \in G $
 +
in such a way that  $  \gamma u ( b) = b $
 +
and  $  u ( 1) = 1 $.  
 +
Then conjugation by  $  u ( b) $
 +
determines an automorphism  $  \phi ( b) $
 +
of  $  A $,
  
<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/e/e036/e036980/e03698050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
 +
\phi ( b) a  = \
 +
u ( b) a u ( b)  ^ {- 1}  = \
 +
{}  ^ {b} a .
 +
$$
  
where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698051.png" /> is implicit in (3).
+
The product of  $  u ( b _ {1} ) $
 +
and  $  u ( b _ {2} ) $
 +
is equal to  $  u ( b _ {1} b _ {2} ) $
 +
up to a factor  $  f ( b _ {1} , b _ {2} ) \in A $:
  
Given groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698053.png" /> and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698055.png" /> satisfying (2), (3) and the normalization conditions
+
$$
 +
u ( b _ {1} ) u ( b _ {2} )  = \
 +
f ( b _ {1} , b _ {2} )
 +
u ( b _ {1} b _ {2} ).
 +
$$
  
<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/e/e036/e036980/e03698056.png" /></td> </tr></table>
+
It is easily checked that these functions must satisfy the conditions
  
one can define an extension (1) in the following way. The product set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698057.png" /> is a group under the operation
+
$$ \tag{2 }
 +
[ \phi ( b _ {1} )
 +
f ( b _ {2} , b _ {3} )]
 +
f ( b _ {1} , b _ {2} b _ {3} )  = \
 +
f ( b _ {1} , b _ {2} )
 +
f ( b _ {1} b _ {2} , b _ {3} ),
 +
$$
  
<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/e/e036/e036980/e03698058.png" /></td> </tr></table>
+
$$ \tag{3 }
 +
{} ^ {b _ {1} } ( {} ^ {b _ {2} } a )  = {} ^ {f (
 +
b _ {1} , b _ {2} ) } ( {} ^ {b _ {1} b _ {2} } a ) ,
 +
$$
  
The homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698060.png" /> yield an extension.
+
where the function  $  \phi : B \rightarrow  \mathop{\rm Aut}  A $
 +
is implicit in (3).
  
Given an abstract kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698061.png" />, it is always possible to find a normalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698062.png" /> satisfying condition (3). A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698063.png" /> arises naturally, but condition (2) is not always fulfilled. In general,
+
Given groups  $  A $
 +
and  $  B $
 +
and functions  $  f: B \times B \rightarrow A $,  
 +
$  \phi : B \rightarrow  \mathop{\rm Aut}  A $
 +
satisfying (2), (3) and the normalization conditions
  
<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/e/e036/e036980/e03698064.png" /></td> </tr></table>
+
$$
 +
\phi ( 1)  = 1,\ \
 +
f ( a, 1)  = = f ( 1, b),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698065.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698066.png" /> is called a factor set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698067.png" /> is called the obstruction to the extension. If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698068.png" /> is Abelian, then the factor sets form a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698069.png" /> under natural composition. Factor sets corresponding to a semi-direct product form a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698070.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698071.png" />. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698072.png" /> is isomorphic to the second cohomology group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698073.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036980/e03698074.png" />. Obstructions have a similar interpretation in the third cohomology group.
+
one can define an extension (1) in the following way. The product set $  A \times B $
 +
is a group under the operation
  
The idea of studying extensions by means of factor sets appeared long ago (O. Hölder, 1893). However, the introduction of factor sets is usually connected with the name of O. Schreier, who used them to undertake the first systematic study of extensions. R. Baer was the first to carry out an invariant study of group extensions without using factor sets. The theory of group extensions is one of the cornerstones of [[Homological algebra|homological algebra]].
+
$$
 +
( a, b) ( a _ {1} , b _ {1} ) = \
 +
( a {}  ^ {b} a _ {1} f ( b, b _ {1} ), bb _ {1} ).
 +
$$
  
====References====
+
The homomorphisms $ a \mapsto ( a, 1) $,  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>
+
$ ( a, b) \mapsto b $
 +
yield an extension.
  
 +
Given an abstract kernel  $  ( A, B, \beta ) $,
 +
it is always possible to find a normalized function  $  \phi $
 +
satisfying condition (3). A function  $  f $
 +
arises naturally, but condition (2) is not always fulfilled. In general,
  
 +
$$
 +
f ( b _ {2} , b _ {3} )
 +
f ( b _ {1} , b _ {2} b _ {3} )  = \
 +
k ( b _ {1} , b _ {2} , b _ {3} )
 +
f ( b _ {1} , b _ {2} )
 +
f ( b _ {1} b _ {2} , b _ {3} ),
 +
$$
  
====Comments====
+
where  $  k ( b _ {1} , b _ {2} , b _ {3} ) \in A $.
 +
The function  $  f:  B \times B \rightarrow A $
 +
is called a factor set and  $  k:  B \times B \times B \rightarrow A $
 +
is called the obstruction to the extension. If the group  $  A $
 +
is Abelian, then the factor sets form a group  $  Z _ {2} ( B, A) $
 +
under natural composition. Factor sets corresponding to a semi-direct product form a subgroup  $  B _ {2} ( B, A) $
 +
of  $  Z _ {2} ( B, A) $.
 +
The quotient group  $  Z _ {2} ( B, A)/B _ {2} ( B, A) $
 +
is isomorphic to the second cohomology group of  $  B $
 +
with coefficients in  $  A $.
 +
Obstructions have a similar interpretation in the third cohomology group.
  
 +
The idea of studying extensions by means of factor sets appeared long ago (O. Hölder, 1893). However, the introduction of factor sets is usually connected with the name of O. Schreier, who used them to undertake the first systematic study of extensions. R. Baer was the first to carry out an invariant study of group extensions without using factor sets. The theory of group extensions is one of the cornerstones of [[Homological algebra|homological algebra]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Eilenberg,  S. MacLane,  "Cohomology theory in abstract groups II"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 326–341</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Eilenberg,  S. MacLane,  "Cohomology theory in abstract groups II"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 326–341</TD></TR></table>

Latest revision as of 20:29, 17 January 2024


A group containing the given group as a normal subgroup. The quotient group is usually prescribed as well, that is, an extension of a group $ A $ by a group $ B $ is a group $ G $ containing $ A $ as a normal subgroup and such that $ G/A \cong B $, i.e. it is an exact sequence

$$ \tag{1 } e \rightarrow A \rightarrow G \mathop \rightarrow \limits ^ \gamma B \rightarrow e. $$

In the literature other terminology is sometimes adopted, e.g., $ G $ may be called an extension of $ B $ by $ A $( see [2], for example), the epimorphism $ \gamma : G \rightarrow B $ itself may be called an extension of $ B $( see [1]), or the exact sequence (1) may be called an extension of $ A $ by $ B $, or an extension of $ B $ by $ A $. An extension of $ A $ by $ B $ always exists, although it is not uniquely determined by $ A $ and $ B $. The need to describe all extensions of $ A $ by $ B $ up to a natural equivalence is motivated by the demands both of group theory itself and of its applications. Two extensions of $ A $ by $ B $ are called equivalent if there is a commutative diagram

$$ \begin{array}{c} e \rightarrow \\ {} \\ e \rightarrow \end{array} \begin{array}{l} A \rightarrow \\ \| \\ A \rightarrow \end{array} \begin{array}{l} G \rightarrow \\ \downarrow \\ G ^ \prime \rightarrow \end{array} \begin{array}{l} B \rightarrow \\ \| \\ B \rightarrow \end{array} \begin{array}{l} e \\ {} \\ e \end{array} $$

Any extension of the form (1) determines, via conjugation of the elements of the group $ G $, a homomorphism $ \alpha : G \rightarrow \mathop{\rm Aut} A $, where $ \mathop{\rm Aut} A $ is the automorphism group of $ A $,

$$ \alpha(g) a = g a g^{-1} , $$

such that $ \alpha ( A) $ is contained in the group $ \mathop{\rm Inn} A $ of inner automorphisms of $ A $. Hence $ \alpha $ induces a homomorphism

$$ \beta : B \rightarrow \ \mathop{\rm Aut} A / \mathop{\rm Inn} A. $$

The triple $ ( A, B, \beta ) $ is called the abstract kernel of the extension. Given an extension (1), one chooses for every $ b \in B $ a representative $ u ( b) \in G $ in such a way that $ \gamma u ( b) = b $ and $ u ( 1) = 1 $. Then conjugation by $ u ( b) $ determines an automorphism $ \phi ( b) $ of $ A $,

$$ \phi ( b) a = \ u ( b) a u ( b) ^ {- 1} = \ {} ^ {b} a . $$

The product of $ u ( b _ {1} ) $ and $ u ( b _ {2} ) $ is equal to $ u ( b _ {1} b _ {2} ) $ up to a factor $ f ( b _ {1} , b _ {2} ) \in A $:

$$ u ( b _ {1} ) u ( b _ {2} ) = \ f ( b _ {1} , b _ {2} ) u ( b _ {1} b _ {2} ). $$

It is easily checked that these functions must satisfy the conditions

$$ \tag{2 } [ \phi ( b _ {1} ) f ( b _ {2} , b _ {3} )] f ( b _ {1} , b _ {2} b _ {3} ) = \ f ( b _ {1} , b _ {2} ) f ( b _ {1} b _ {2} , b _ {3} ), $$

$$ \tag{3 } {} ^ {b _ {1} } ( {} ^ {b _ {2} } a ) = {} ^ {f ( b _ {1} , b _ {2} ) } ( {} ^ {b _ {1} b _ {2} } a ) , $$

where the function $ \phi : B \rightarrow \mathop{\rm Aut} A $ is implicit in (3).

Given groups $ A $ and $ B $ and functions $ f: B \times B \rightarrow A $, $ \phi : B \rightarrow \mathop{\rm Aut} A $ satisfying (2), (3) and the normalization conditions

$$ \phi ( 1) = 1,\ \ f ( a, 1) = 1 = f ( 1, b), $$

one can define an extension (1) in the following way. The product set $ A \times B $ is a group under the operation

$$ ( a, b) ( a _ {1} , b _ {1} ) = \ ( a {} ^ {b} a _ {1} f ( b, b _ {1} ), bb _ {1} ). $$

The homomorphisms $ a \mapsto ( a, 1) $, $ ( a, b) \mapsto b $ yield an extension.

Given an abstract kernel $ ( A, B, \beta ) $, it is always possible to find a normalized function $ \phi $ satisfying condition (3). A function $ f $ arises naturally, but condition (2) is not always fulfilled. In general,

$$ f ( b _ {2} , b _ {3} ) f ( b _ {1} , b _ {2} b _ {3} ) = \ k ( b _ {1} , b _ {2} , b _ {3} ) f ( b _ {1} , b _ {2} ) f ( b _ {1} b _ {2} , b _ {3} ), $$

where $ k ( b _ {1} , b _ {2} , b _ {3} ) \in A $. The function $ f: B \times B \rightarrow A $ is called a factor set and $ k: B \times B \times B \rightarrow A $ is called the obstruction to the extension. If the group $ A $ is Abelian, then the factor sets form a group $ Z _ {2} ( B, A) $ under natural composition. Factor sets corresponding to a semi-direct product form a subgroup $ B _ {2} ( B, A) $ of $ Z _ {2} ( B, A) $. The quotient group $ Z _ {2} ( B, A)/B _ {2} ( B, A) $ is isomorphic to the second cohomology group of $ B $ with coefficients in $ A $. Obstructions have a similar interpretation in the third cohomology group.

The idea of studying extensions by means of factor sets appeared long ago (O. Hölder, 1893). However, the introduction of factor sets is usually connected with the name of O. Schreier, who used them to undertake the first systematic study of extensions. R. Baer was the first to carry out an invariant study of group extensions without using factor sets. The theory of group extensions is one of the cornerstones of homological algebra.

References

[1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[3] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[4] S. MacLane, "Homology" , Springer (1963)
[a1] S. Eilenberg, S. MacLane, "Cohomology theory in abstract groups II" Ann. of Math. , 48 (1947) pp. 326–341
How to Cite This Entry:
Extension of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_group&oldid=13641
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article