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) |
Comments
References
| [a1] | S. Eilenberg, S. MacLane, "Cohomology theory in abstract groups II" Ann. of Math. , 48 (1947) pp. 326–341 |
Extension of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_a_group&oldid=46880