Monomial Group
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In mathematics, in the area of
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters 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 power product, is a product of powers of variables with nonnegative integer expon ...
, 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 See also * Inducement (disambiguation ...
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 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 ...
and every solvable
A-group The A-Group culture was an ancient culture that flourished between the First and Second Cataracts of the Nile in Nubia. It lasted from 3800 BC to 3100 BC. Overview In 1907, the Egyptologist George A. Reisner first discovered artifacts belongi ...
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 group ...
S_4 is an example of a monomial group that is neither supersolvable nor an
A-group The A-Group culture was an ancient culture that flourished between the First and Second Cataracts of the Nile in Nubia. It lasted from 3800 BC to 3100 BC. Overview In 1907, the Egyptologist George A. Reisner first discovered artifacts belongi ...
. The
special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the gen ...
\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.


References

* * * Finite groups Properties of groups {{Abstract-algebra-stub