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 ...

studying the

character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information a ...

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

s, an M-group or monomial group is a finite

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 ...

whose complex irreducible

characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to Theoph ...

are all

monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...

, that is,

induced Induce may refer to: * Induced consumption * Induced innovation * Induced character * Induced coma * Induced menopause * Induced metric * Induced path * Induced topology * Induce (musician), American musician * Labor induction, stimulation of chil ...

from characters of degree 1. In this section only finite groups are considered. A monomial group is solvable. Every

supersolvable group In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability. Definition Let ''G'' be a group. ''G'' is sup ...

and every solvable

A-group The A-Group was the first powerful society in Nubia, located in modern northern Sudan and southern Egypt and flourished between the First and Second Cataracts of the Nile in Lower Nubia. It lasted from the 4th millennium BC, reached its climax ...

is a monomial group. Factor groups of monomial groups are monomial, but subgroups need not be, since every finite solvable group can be embedded in a monomial group.As shown by and in textbook form in . The

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

S_4

is an example of a monomial group that is neither supersolvable nor an

. The

special linear group In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...

\operatorname_2(\mathbb F_3)

is the smallest finite group that is not monomial: since the abelianization of this group has order three, its irreducible characters of degree two are not monomial.

Notes

References

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