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 by a group is a group containing as a normal subgroup and such that , i.e. it is an exact sequence

(1)

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

Any extension of the form (1) determines, via conjugation of the elements of the group , a homomorphism , where is the automorphism group of ,

such that is contained in the group of inner automorphisms of . Hence induces a homomorphism

The triple is called the abstract kernel of the extension. Given an extension (1), one chooses for every a representative in such a way that and . Then conjugation by determines an automorphism of ,

The product of and is equal to up to a factor :

It is easily checked that these functions must satisfy the conditions

(2)

(3)

where the function is implicit in (3).

Given groups and and functions , satisfying (2), (3) and the normalization conditions

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

The homomorphisms , yield an extension.

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

where . The function is called a factor set and is called the obstruction to the extension. If the group is Abelian, then the factor sets form a group under natural composition. Factor sets corresponding to a semi-direct product form a subgroup of . The quotient group is isomorphic to the second cohomology group of with coefficients in . 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)

References

[a1]	S. Eilenberg, S. MacLane, "Cohomology theory in abstract groups II" Ann. of Math. , 48 (1947) pp. 326–341