Monomial group
Encyclopedia
In mathematics
, in the area of algebra
studying the character theory
of finite group
s, an M-group or monomial group is a finite group
whose complex irreducible characters
are all monomial, that is, induced
from characters of degree 1 .
In this section only finite groups are considered. A monomial group is solvable
by , presented in textbook in and . Every supersolvable group
and every solvable A-group
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 .
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 area of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
studying the character theory
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
of 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...
s, an M-group or monomial group is a finite 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...
whose complex irreducible characters
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
are all monomial, that is, induced
Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
from characters of degree 1 .
In this section only finite groups are considered. A monomial group is 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...
by , presented in textbook in and . Every supersolvable group
Supersolvable group
In mathematics, a group is supersolvable 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...
and every solvable A-group
A-group
A-Group is the designation for a distinct culture that arose between the First and Second Cataracts of the Nile in Nubia betweenthe Egyptian 1st dynasty and the 3rd millennium BC.The A-Group settled on very poor land with scarce natural resources, yet...
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 .