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.
Read more about this topic: Schur Multiplier
Famous quotes containing the words relation to, relation, central and/or extensions:
“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)
“Parents ought, through their own behavior and the values by which they live, to provide direction for their children. But they need to rid themselves of the idea that there are surefire methods which, when well applied, will produce certain predictable results. Whatever we do with and for our children ought to flow from our understanding of and our feelings for the particular situation and the relation we wish to exist between us and our child.”
—Bruno Bettelheim (20th century)
“The fantasies inspired by TB in the last century, by cancer now, are responses to a disease thought to be intractable and capricious—that is, a disease not understood—in an era in which medicine’s central premise is that all diseases can be cured.”
—Susan Sontag (b. 1933)
“The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we don’t like what we see, we feel angry at the reflection.”
—Elaine Heffner (20th century)