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.{{sfnp|Isaacs|1994}}
In this section only finite groups are considered. A monomial group is solvable.By {{harv|Taketa|1930}}, presented in textbook in {{harv|Isaacs|1994|loc=Cor. 5.13}} and {{harv|Bray|Deskins|Johnson|Humphreys|1982|loc=Cor 2.3.4}}. Every supersolvable group{{sfnp|Bray|Deskins|Johnson|Humphreys|1982|loc=Cor 2.3.5}} and every solvable A-group{{sfnp|Bray|Deskins|Johnson|Humphreys|1982|loc=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 {{harv|Dade|1988}} and in textbook form in {{harv|Bray|Deskins|Johnson|Humphreys|1982|loc=Ch 2.4}}.
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
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References
- {{Citation | last1=Bray | first1=Henry G. | last2=Deskins | first2=W. E. | last3=Johnson | first3=David | last4=Humphreys | first4=John F. | last5=Puttaswamaiah | first5=B. M. | last6=Venzke | first6=Paul | last7=Walls | first7=Gary L. | title=Between nilpotent and solvable | publisher=Polygonal Publ. House | location=Washington, N. J. | isbn=978-0-936428-06-2 |mr=655785 | year=1982}}
- {{citation
| last = Dade | first = Everett C.
| doi = 10.1016/0021-8693(88)90253-0
| issue = 1
| journal = Journal of Algebra
| mr = 955603
| pages = 256–266
| title = Accessible characters are monomial
| volume = 117
| year = 1988}}
- {{Citation | last1=Isaacs | first1=I. Martin | title=Character Theory of Finite Groups | publisher=Dover Publications | location=New York | isbn=978-0-486-68014-9 | year=1994}}
- {{Citation | last1=Taketa | first1=K. | title=Über die Gruppen, deren Darstellungen sich sämtlich auf monomiale Gestalt transformieren lassen. | language=German | year=1930 | journal=Proceedings of the Imperial Academy | volume=6 | pages=31–33 | doi=10.3792/pia/1195581421 | issue=2| doi-access=free }}
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