In mathematics, especially in the field of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, a Carter subgroup of a finite group ''G'' is a

self-normalizing subgroup In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...

of ''G'' that is

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...

. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of

solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group (mathematics), group that can be constructed from abelian groups using Group extension, extensions. Equivalently, a solvable group is a ...

s . proved that any finite

has a Carter subgroup, and all its Carter subgroups are

conjugate subgroup In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other w ...

s (and therefore isomorphic). If a group is not solvable it need not have any Carter subgroups: for example, the

alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...

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 In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centra ...

. viewed the Carter subgroups as analogues of

Sylow subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...

s and

Hall subgroup In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist . Definitions A Hall divisor (also called a unitary divisor) of ...

s, and unified their treatment with the theory of formations. In the language of formations, a Sylow ''p''-subgroup is a 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 .

References

* * * *, especially Kap VI, §12, pp736–743 * * * translation in

Siberian Mathematical Journal The ''Siberian Mathematical Journal'' (abbreviated as Sib. Math. J.) is a cover-to-cover English translation of the Russian peer-reviewed mathematics journal ''Sibirskii Matematicheskii Zhurnal'', a publication of the Sobolev Institute of Mathem ...

47 (2006), no. 4, 597–600. * * * Finite groups Solvable groups Subgroup properties {{Abstract-algebra-stub

See also

References