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:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
—William Harmon (b. 1938)
“In inner-party politics, these methods lead, as we shall yet see, to this: the party organization substitutes itself for the party, the central committee substitutes itself for the organization, and, finally, a “dictator” substitutes himself for the central committee.”
—Leon Trotsky (1879–1940)
“If we focus exclusively on teaching our children to read, write, spell, and count in their first years of life, we turn our homes into extensions of school and turn bringing up a child into an exercise in curriculum development. We should be parents first and teachers of academic skills second.”
—Neil Kurshan (20th century)