Extension of a group

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