Monomial group

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 ${\displaystyle S_{4}}$ is an example of a monomial group that is neither supersolvable nor an A-group. The special linear group ${\displaystyle \operatorname {SL} _{2}(\mathbb {F} _{3})}$ 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.