Monomial groups

A finite group G is monomial (an M-group) when every irreducible complex character of G is induced from a degree-1 character of a subgroup. Monomial groups sit strictly between supersolvable and solvable: every supersolvable group is monomial, and every monomial group is solvable — but neither converse holds.

Definition

A monomial character of G is a character induced from a linear (degree-1) character of a subgroup:

χ = λᴳ,   λ : H → ℂˣ a homomorphism,   H ≤ G,   deg χ = [G : H].

G is monomial when every irreducible complex character of G is monomial.

Equivalently, every irreducible complex representation of G can be realized by monomial matrices — invertible matrices with exactly one nonzero entry in each row and each column (a permutation matrix times an invertible diagonal matrix), such as

⎡ 0  a  0 ⎤
⎢ 0  0  b ⎥ .
⎣ c  0  0 ⎦

A representation whose image consists of monomial matrices is precisely one induced from a 1-dimensional representation of a subgroup — the origin of the name.

Placement

Among finite groups the inclusions

abelian ⊂ nilpotent ⊂ supersolvable ⊂ monomial ⊂ solvable

hold, and every inclusion is strict.

• Supersolvable ⇒ monomial. In a supersolvable group every irreducible character is monomial. The converse fails.
• Taketa's theorem: monomial ⇒ solvable. Every M-group is solvable. The converse fails.

Witnesses of the two strict boundaries:

Properties

• Taketa's inequality. For an M-group the derived length is bounded by the number of distinct irreducible character degrees: dl(G) ≤ |cd(G)|. (Whether this bound holds for every finite solvable group — the Taketa problem — is an open problem.)
• Closed under quotients. Every quotient of an M-group is an M-group.
• Closed under finite direct products. G × H is monomial whenever G and H are.
• Not closed under subgroups. An M-group can contain a subgroup that is not monomial. This failure is sharp: Dade (1973) constructed an M-group with a non-monomial normal subgroup, so closure fails even for normal subgroups.

Examples

• Every supersolvable group — including every abelian group, every nilpotent group (hence every p-group), every dihedral group Dₙ, and every metacyclic group.
• S₄ — solvable and monomial but not supersolvable (its maximal subgroup S₃ has index 4, not prime). Its irreducible degrees are 1, 1, 2, 3, 3, and each irreducible is induced from a linear character of a subgroup.
• A₄ — order 12, degrees 1, 1, 1, 3. The degree-3 character is induced from a linear character of the index-3 subgroup V₄ = C₂ × C₂.

Non-examples

• SL(2,𝔽3) — order 24, the smallest non-monomial group. It is solvable (commutator subgroup Q₈, abelianization C₃), but having no subgroup of index 2 it can have no monomial character of degree 2; its three degree-2 irreducible characters are therefore not monomial.
• A₅ — simple and non-abelian, hence not solvable, hence not monomial by Taketa's theorem.

See also

• Supersolvable group — every supersolvable group is monomial.
• Solvable group — every monomial group is solvable (Taketa).
• Conjugacy classes and characters — induced characters and the character-theory background.
• Perfect group — a nontrivial perfect group is never monomial (it is not solvable).
• Matrix groups — where SL(2,𝔽3) and A₅ live.
• Wikipedia: Monomial group