Implications and limits for M-groups

For finite groups the implications

$\text{supersolvable}⟹M\text{-group}⟹\text{solvable}$

hold (Monomial representations, monomial characters, and M-groups, Finite supersolvable groups are M-groups, M-groups are solvable (Taketa)).

• Strictness of the second implication. The companion counterexample A solvable group that is not an M-group ↗ constructs the binary tetrahedral group $T=Q_8⋊C_3$ of order $24$ and verifies that it is solvable, has a faithful irreducible complex representation of degree two, and has no subgroup of index two. The dimension formula The dimension of an induced finite-dimensional representation is $[G:H]\dim W$ then excludes induction of that representation from a linear character, so the later example proves that solvable does not imply $M$-group. These group facts are established there, not used to derive the implication chain above.

• Brauer's monomial induction theorem (Virtual characters are integrally generated by monomial characters from elementary subgroups) is a different assertion from "$G$ is an $M$-group": it states that every virtual character of $G$ is an integral combination of monomial characters induced from elementary subgroups, with coefficients that may be negative and with elementary subgroups that need not be the inertia groups of the irreducible constituents. It therefore does not produce a monomial irreducible character of $G$, and the companion counterexample shows that it cannot: for $T=Q_8⋊C_3$ the virtual-character statement holds by Virtual characters are integrally generated by monomial characters from elementary subgroups while one of the irreducible characters of $T$ is not monomial.

• This remark does not address whether the first implication reverses. The standard witness that not every $M$-group is supersolvable is not developed on this page.

• The implication chain above is the reason this page treats supersolvable groups as the principal source of $M$-groups: combining Finite supersolvable groups are M-groups with M-groups are solvable (Taketa) and The derived series, solvable groups, and derived length places every finite supersolvable group in the hierarchy supersolvable $\Rightarrow$ $M$-group $\Rightarrow$ solvable, in which the second inclusion is strict by the companion counterexample and the first is not settled here.