In mathematics, in the area of algebra studying the character theory of finite groups, 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.[1]
In this section only finite groups are considered. A monomial group is solvable.[2] Every supersolvable group[3] and every solvable A-group[4] 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.[5]
The symmetric group is an example of a monomial group that is neither supersolvable nor an A-group. The special linear group 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.
^Isaacs (1994).
^By (Taketa 1930), presented in textbook in (Isaacs 1994, Cor. 5.13) and (Bray et al. 1982, Cor 2.3.4).
^Bray et al. (1982), Cor 2.3.5.
^Bray et al. (1982), Thm 2.3.10.
^As shown by (Dade 1988) and in textbook form in (Bray et al. 1982, Ch 2.4).
character theory of finite groups, an M-group or monomialgroup is a finite group whose complex irreducible characters are all monomial, that is, induced from...
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...
finite supersolvable group is a monomialgroup. Every maximal subgroup in a supersolvable group has prime index. A finite group is supersolvable if and...
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is...
representation theory and group theory, a linear representation ρ {\displaystyle \rho } (rho) of a group G {\displaystyle G} is a monomial representation if there...
standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving...
polynomials that contain only one type of monomial, with only those copies required to obtain symmetry. Any monomial in X1, ..., Xn can be written as X1α1...
suspected that Co0 was transitive on Λ2, and indeed he found a new matrix, not monomial and not an integer matrix. Let η be the 4-by-4 matrix 1 2 ( 1 − 1 − 1 −...
a monomial does not change the order of the terms. For Gröbner bases, a further condition must be satisfied, namely that every non-constant monomial is...
order m. As a matrix group, its elements may be realized as monomial matrices whose nonzero elements are mth roots of unity. The group G(m, p, n) is an index-p...
[ X , X − 1 ] {\displaystyle R[X,X^{-1}]} is isomorphic to the group ring of the group Z {\displaystyle \mathbb {Z} } of integers over R {\displaystyle...
size i. The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also...
their irreducible representations are monomial, the so-called monomialgroups. If H is a subgroup of the group G, then every K-linear representation ρ...
even non-negative, as positive matrices are not monomial, for dimension n > 1. There are a number of groups of matrices that form specializations of non-negative...
in J (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the...
represented by monomial matrices (matrices having one nonzero entry in every row and column) whose nonzero entries are all mth roots of unity. The groups G ( m...
of η is odd. Another representation fixes the vector v = (4,-4,022). A monomial and maximal subgroup includes a representation of M22:2, where any α interchanging...
Poincaré–Birkhoff–Witt theorem, it is thus the free vector space generated by the monomials z j p 1 k 1 p 2 k 2 ⋯ p n k n q 1 ℓ 1 q 2 ℓ 2 ⋯ q n ℓ n , {\displaystyle...
each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates...
edge x-y = (1, 5, 122) is type 3; it is fixed by a Co3. This M22 is the monomial, and a maximal, subgroup of a representation of McL. Wilson (2009) (p. 207)...
extends to monomials; thus, sin 3x = sin(3x) and even sin 1/2xy = sin(xy/2), but sin x + y = sin(x) + y, because x + y is not a monomial. However, this...
respective degrees 0, 1, …, n. Their matrix on the monomial basis is an upper-triangular matrix U (if the monomials are ordered in increasing degrees), with all...
GL(K), consisting of monomial matrices, and BGM(K)+ is the Quillen's plus-construction. Moreover, let K3M denote the Milnor's K-group, then there exists...
may be written as a sum of terms, each consisting of an integral over a monomial of coordinates x i {\displaystyle x_{i}} and fields ϕ i {\displaystyle...
Global Information
