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) and (Bray Deskins). Every supersolvable group (Bray Deskins) and every solvable A-group (Bray Deskins) 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 Deskins).
The symmetric group [math]\displaystyle{ S_4 }[/math] is an example of a monomial group that is neither supersolvable nor an A-group. The special linear group [math]\displaystyle{ \operatorname{SL}_2(\mathbb F_3) }[/math] 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
- 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." (in German), Proceedings of the Imperial Academy 6 (2): 31–33, doi:10.3792/pia/1195581421
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