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 (Isaacs 1994).

In this section only finite groups are considered. A monomial group is solvable by (Taketa 1930), presented in textbook in (Isaacs 1994, Cor. 5.13) and (Bray et al. 1982, Cor 2.3.4). Every supersolvable group (Bray et al. 1982, Cor 2.3.5) and every solvable A-group (Bray et al. 1982, Thm 2.3.10) 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 (Dade & ????) and in textbook form in (Bray et al. 1982, Ch 2.4).

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

References

• Bray, Henry G.; Deskins, W. E.; Johnson, David; Humphreys, John F.; Puttaswamaiah, B. M.; Venzke, Paul; Walls, Gary L. (1982), Between nilpotent and solvable, Washington, N. J.: Polygonal Publ. House, ISBN 978-0-936428-06-2, MR 0655785
• Isaacs, I. Martin (1994), Character Theory of Finite Groups, New York: Dover Publications, ISBN 978-0-486-68014-9
• Taketa, K. (1930), "Über die Gruppen, deren Darstellungen sich sämtlich auf monomiale Gestalt transformieren lassen.", Proceedings of the Imperial Academy (in German), 6 (2): 31–33, doi:10.3792/pia/1195581421