mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in the area of

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

known as

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 more than fifty-year effort was made to answer a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

of : are all

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

s of odd

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

solvable? Progress was made by showing that CA-groups, groups in which the

centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

of a non-

identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an ...

element is

abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group ...

, of odd order are solvable . Further progress was made showing that CN-groups, groups in which the centralizer of a non-identity element is

nilpotent In mathematics, an element x of a ring (mathematics), 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, along with its sister Idempotent (ring theory), idem ...

, of odd order are solvable . The complete solution was given in , but further work on CN-groups was done in , giving more detailed information about the structure of these groups. For instance, a non-solvable CN-group ''G'' is such that its largest solvable normal subgroup ''O''_∞(''G'') is a

2-group In mathematics, particularly category theory, a is a groupoid with a way to multiply objects and morphisms, making it resemble a group. They are part of a larger hierarchy of . They were introduced by Hoàng Xuân Sính in the late 1960s unde ...

, and the

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

is a group of

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname), a Breton surname * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a ...

order.

Examples

Solvable CN groups include *Nilpotent groups *

Frobenius group In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius. Structure Suppos ...

s whose Frobenius complement is nilpotent *

3-step group In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different. C ...

s, such as the symmetric group ''S''₄ Non-solvable CN groups include: *The Suzuki simple groups *The groups PSL₂(F_2^''n'') for ''n''>1 *The group PSL₂(F_''p'') for ''p''>3 a

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...

. *The group PSL₂(F₉) *The group PSL₃(F₄)

References

* * * * * Finite groups Group theory Properties of groups {{group-theory-stub