CA group
Encyclopedia
In mathematics
, in the realm of group theory
, a group
is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian
subgroup
. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theorem
and the classification of finite simple groups
. Several important infinite groups are CA-groups, such as free group
s, Tarski monsters, and some Burnside groups, and the locally finite
CA-groups have been classified explicitly. CA-groups are also called commutative-transitive groups (or CT-groups for short) because commutativity is a transitive relation
amongst the non-identity elements of a group if and only if the group is a CA-group.
or solvable
in . Then in the Brauer-Suzuki-Wall theorem , finite CA-groups of even order were shown to be Frobenius group
s, abelian groups, or two dimensional projective special linear groups over a finite field
of even order, PSL(2,2f) for f ≥ 2. Finally, finite CA-groups of odd order were shown to be Frobenius group
s or abelian groups in , and so in particular, are never non-abelian simple.
CA-groups were important in the context of the classification of finite simple groups
. Michio Suzuki
showed that every finite
, simple
, non-abelian, CA-group is of even order
. This result was first extended to the Feit–Hall–Thompson theorem showing that finite, simple, non-abelian, CN-groups had even order, and then to the Feit–Thompson theorem
which states that every finite, simple, non-abelian group is of even order. A textbook exposition of the classification of finite CA-groups is given as example 1 and 2 in . A more detailed description of the Frobenius groups appearing is included in , where it is shown that a finite, solvable CA-group is a semidirect product
of an abelian group and a fixed-point-free automorphism, and that conversely every such semidirect product is a finite, solvable CA-group. Wu also extended the classification of Suzuki et al. to locally finite group
s.
is a CA-group, and a group with a non-trivial center
is a CA-group if and only if it is abelian. The finite CA-groups are classified, and include groups such as the dihedral group
s of order 4k+2, and the alternating group on 4 points of order 12. Infinite CA-groups include free group
s, PSL2(R), and Burnside groups of large prime exponent, . Some more recent results in the infinite case are included in , including a classification of locally finite
CA-groups. Wu also observes that Tarski monsters are obvious examples of infinite simple CA-groups.
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...
, in the realm 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 group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
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...
. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in the Feit–Thompson theorem
Feit–Thompson theorem
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...
and the classification of finite simple groups
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
. Several important infinite groups are CA-groups, such as free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s, Tarski monsters, and some Burnside groups, and the locally finite
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 studied....
CA-groups have been classified explicitly. CA-groups are also called commutative-transitive groups (or CT-groups for short) because commutativity is a transitive relation
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
amongst the non-identity elements of a group if and only if the group is a CA-group.
History
Locally finite CA-groups were classified by several mathematicians from 1925 to 1998. First, finite CA-groups were shown to be simpleSimple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
or solvable
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...
in . Then in the Brauer-Suzuki-Wall theorem , finite CA-groups of even order were shown to be Frobenius group
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point.They are named after F. G. Frobenius.- Structure :...
s, abelian groups, or two dimensional projective special linear groups over a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
of even order, PSL(2,2f) for f ≥ 2. Finally, finite CA-groups of odd order were shown to be Frobenius group
Frobenius group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point.They are named after F. G. Frobenius.- Structure :...
s or abelian groups in , and so in particular, are never non-abelian simple.
CA-groups were important in the context of the classification of finite simple groups
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
. Michio Suzuki
Michio Suzuki
was a Japanese mathematician who studied group theory.-Biography:He was a Professor at the University of Illinois at Urbana-Champaign from 1953 to his death. He also had visiting positions at the University of Chicago , the Institute for Advanced Study , the University of Tokyo , and the...
showed that every finite
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...
, simple
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
, non-abelian, CA-group is of even order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
. This result was first extended to the Feit–Hall–Thompson theorem showing that finite, simple, non-abelian, CN-groups had even order, and then to the Feit–Thompson theorem
Feit–Thompson theorem
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...
which states that every finite, simple, non-abelian group is of even order. A textbook exposition of the classification of finite CA-groups is given as example 1 and 2 in . A more detailed description of the Frobenius groups appearing is included in , where it is shown that a finite, solvable CA-group is a semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of an abelian group and a fixed-point-free automorphism, and that conversely every such semidirect product is a finite, solvable CA-group. Wu also extended the classification of Suzuki et al. to locally finite group
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 studied....
s.
Examples
Every abelian groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
is a CA-group, and a group with a non-trivial center
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
is a CA-group if and only if it is abelian. The finite CA-groups are classified, and include groups such as the dihedral group
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
s of order 4k+2, and the alternating group on 4 points of order 12. Infinite CA-groups include free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s, PSL2(R), and Burnside groups of large prime exponent, . Some more recent results in the infinite case are included in , including a classification of locally finite
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 studied....
CA-groups. Wu also observes that Tarski monsters are obvious examples of infinite simple CA-groups.