Carter subgroup

In mathematics, especially in the field of group theory, a Carter subgroup of a finite group "G" is a subgroup "H" that is a nilpotent group, and self-normalizing. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups harv|Wehrfritz|1999.

harvtxt|Carter|1961 proved that any finite solvable group has a Carter subgroup, and all its Carter subgroups are conjugate subgroups (and therefore isomorphic). If a group is not solvable it need not have any Carter subgroups: for example, the alternating group A₅ of order 60 has no Carter subgroups. harvs|txt=yes|last=Vdovin|year=2006|year2=2007 showed that even if a finite group is not solvable then any two Carter subgroups are conjugate.

A Carter subgroup is a maximal nilpotent subgroup, because of the normalizer condition for nilpotent groups, but not all maximal nilpotent subgroups are Carter subgroups harv | Ballester-Bolinches | Ezquerro | 2006 | p=100 . For example, any non-identity proper subgroup of the nonabelian group of order six is a maximal nilpotent subgroup, but only those of order two are Carter subgroups. Every subgroup containing a Carter subgroup of a soluble group is also self-normalizing, and a soluble group is generated by any Carter subgroup and its nilpotent residual harv|Schenkman|1975|loc=VII.4.a.

harv | Gaschütz | 1963 viewed the Carter subgroups as analogues of Sylow subgroups and Hall subgroups, and unified their treatment with the theory of formations. In the language of formations, a Sylow "p"-subgroup is covering group for the formation of "p"-groups, a Hall "π"-subgroup is a covering group for the formation of "π"-groups, and a Carter subgroup is a covering group for the formation of nilpotent groups harv | Ballester-Bolinches | Ezquerro | 2006 | p=100 . Together with an important generalization, Schunck classes, and an important dualization, Fischer classes, formations formed the major research themes of the late 20th century in the theory of finite soluble groups.

A dual notion to Carter subgroups was introduced by Bernd Fischer in harv|Fischer|1966. A Fischer subgroup of a group is a nilpotent subgroup containing every other nilpotent subgroup it normalizes. A Fischer subgroup is a maximal nilpotent subgroup, but not every maximal nilpotent subgroup is a Fischer subgroup: again the nonabelian group of order six provides an example as every non-identity proper subgroup is a maximal nilpotent subgroup, but only the subgroup of order three is a Fischer subgroup harv|Wehrfritz|1999|p=98.

ee also

*Cartan subalgebra
*Cartan subgroup

References

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

Cartan subgroup — In mathematics, a Cartan subgroup of a Lie group or algebraic group G is one of the subgroups whose Lie algebrais a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the rank of G .ConventionsThe… … Wikipedia
Roger Carter (mathematician) — Roger W. Carter is an emeritus professor at University of Warwick. He defined Carter subgroups and wrote the standard reference Simple Groups of Lie Type . He obtained his PhD in 1960 and his dissertation was entitled Some Contributions to the… … Wikipedia
A-group — In mathematics, in the area of abstract algebra known as group theory, an A group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known… … Wikipedia
Locally finite group — In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been… … Wikipedia
List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… … Wikipedia
Cartan subalgebra — In mathematics, a Cartan subalgebra is a nilpotent subalgebra mathfrak{h} of a Lie algebra mathfrak{g} that is self normalising (if [X,Y] in mathfrak{h} for all X in mathfrak{h}, then Y in mathfrak{h}).Cartan subalgebras exist for finite… … Wikipedia
History of group theory — The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry.… … Wikipedia
Deligne–Lusztig theory — In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ adic cohomology with compact support, introduced by Deligne Lusztig (1976). Lusztig (1984) used these representations to… … Wikipedia
Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines … Wikipedia
Boundedly generated group — In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence subgroup problem (see harvnb|Lubotzky|Segal|2003) … Wikipedia

Academic Dictionaries and Encyclopedias

Carter subgroup

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Carter subgroup

Look at other dictionaries:

Share the article and excerpts

Direct link