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.

Notes

1. Isaacs (1994).
2. By (Taketa 1930), presented in textbook in (Isaacs 1994, Cor. 5.13) and (Bray et al. 1982, Cor 2.3.4).
3. Bray et al. (1982), Cor 2.3.5.
4. Bray et al. (1982), Thm 2.3.10.
5. As shown by (Dade 1988) and in textbook form in (Bray et al. 1982, Ch 2.4).

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
• Dade, Everett C. (1988), "Accessible characters are monomial", Journal of Algebra, 117 (1): 256–266, doi:10.1016/0021-8693(88)90253-0, MR 0955603
• 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