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 is an example of a monomial group that is neither supersolvable nor an A-group. The special linear group 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
[edit]- ^ Isaacs (1994).
- ^ By (Taketa 1930), presented in textbook in (Isaacs 1994, Cor. 5.13) and (Bray et al. 1982, Cor 2.3.4).
- ^ Bray et al. (1982), Cor 2.3.5.
- ^ Bray et al. (1982), Thm 2.3.10.
- ^ As shown by (Dade 1988) and in textbook form in (Bray et al. 1982, Ch 2.4).
References
[edit]- 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
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