Carter subgroup
Encyclopedia
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 group
s .
proved that any finite solvable group
has a Carter subgroup, and all its Carter subgroups are conjugate subgroup
s (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. 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 . 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 .
viewed the Carter subgroups as analogues of Sylow subgroups and Hall subgroup
s, 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 . 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 . 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 .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the field of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a Carter subgroup of a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
G is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
H that is a nilpotent group
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
, and self-normalizing. These subgroups were introduced by Roger Carter
Roger Carter (mathematician)
Roger W. Carter is an emeritus professor at the 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 Theory of Finite Soluble Groups.-Publications:*R.W....
, and marked the beginning of the post 1960 theory of solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
s .
proved that any finite solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
has a Carter subgroup, and all its Carter subgroups are conjugate subgroup
Inner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
s (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. 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 . For example, any non-identity proper subgroup of the nonabelian group of order six
Dihedral group of order 6
The smallest non-abelian group has 6 elements. It is a dihedral group with notation D3 and the symmetric group of degree 3, with notation S3....
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 .
viewed the Carter subgroups as analogues of Sylow subgroups and Hall subgroup
Hall subgroup
In mathematics, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They are named after the group theorist Philip Hall.- Definitions :A Hall divisor of an integer n is a divisor d of n such that...
s, and unified their treatment with the theory of formations
Formation (group theory)
In mathematical group theory, a formation is a class of groups closed under taking images and such that if G/M and G/N are in the formation then so is G/M∩N...
. 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 . 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
Bernd Fischer
Bernd Fischer may refer to:* Bernd Fischer , German mathematician.* Bernd Jürgen Fischer, historian and professor of history at Indiana University-Purdue University Fort Wayne....
in . 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 .