Lesson plan
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. In this section only finite groups are considered. A monomial group is solvable. Every supersolvable group and every solvable A-group 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. 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.
Article summary
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. In this section only finite groups are considered. A monomial group is solvable. Every supersolvable group and every solvable A-group 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. 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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
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