Schur Multiplier - Relation To Central Extensions

Relation To Central Extensions

The study of such covering groups led naturally to the study of central and stem extensions.

A central extension of a group G is an extension

1 → K → C → G → 1

where K ≤ Z(C) is a subgroup of the center of C.

A stem extension of a group G is an extension

1 → K → C → G → 1

where K ≤ Z(C) ∩ C′ is a subgroup of the intersection of the center of C and the derived subgroup of C; this is more restrictive than central.

If the group G is finite and one considers only stem extensions, then there is a largest size for such a group C, and for every C of that size the subgroup K is isomorphic to the Schur multiplier of G. If the finite group G is moreover perfect, then C is unique up to isomorphism and is itself perfect. Such C are often called universal perfect central extensions of G, or covering group (as it is a discrete analog of the universal covering space in topology). If the finite group G is not perfect, then its Schur covering groups (all such C of maximal order) are only isoclinic.

It is also called more briefly a universal central extension, but note that there is no largest central extension, as the direct product of G and an abelian group form a central extension of G of arbitrary size.

Stem extensions have the nice property that any lift of a generating set of G is a generating set of C. If the group G is presented in terms of a free group F on a set of generators, and a normal subgroup R generated by a set of relations on the generators, so that G ≅ F/R, then the covering group itself can be presented in terms of F but with a smaller normal subgroup S, C ≅ F/S. Since the relations of G specify elements of K when considered as part of C, one must have S ≤ .

In fact if G is perfect, this is all that is needed: C ≅ / and M(G) ≅ K ≅ R/. Because of this simplicity, expositions such as (Aschbacher 2000, §33) handle the perfect case first. The general case for the Schur multiplier is similar but ensures the extension is a stem extension by restricting to the derived subgroup of F: M(G) ≅ (R ∩ )/. These are all slightly later results of Schur, who also gave a number of useful criteria for calculating them more explicitly.