nLab Taketa's theorem

Idea

Taketa’s theorem and related statements say that a necessary condition on a finite group to have all its complex irreps be suitably induced representations is that it is a solvable group.

Terminology: “M-groups”

A finite group $G$ is called an $M$-group if all its complex irreducible representations are induced from 1-dimensional representations, hence if for each irrep $rho$ of $G$ there exists a subgroup $H \subset G$ and a 1-dimensional rep $\nu$ of $H$, such that $\rho \,\simeq\, Ind^G_H \rho$, where $Ind_H^G \,\colon\, H Rep \to G Rep$ is the induction functor.

More generally, for $k \in \mathbb{N}_+$ a positive natural number, a finite group is called an $M_k$-group if all its irreps are induced from irreps of dimension $\leq k$. (cf. Berkovich 1996)

Statement

Theorem

(Taketa’s theorem)
M-groups are solvable.

(Taketa 1930, Isaacs 1976 thm. 5.12)

In fact:

Theorem

If a finite group has the property that for each of its complex irreps of dimension $\geq 2$ some multiple of it is induced from an irrep of a proper subgroup, then the group is solvable.

(LMT 2013)

References

Taketa’s theorem is due to

• Kiyosi Taketa: Über die Gruppen, deren Darstellungen sich sämtlich auf monomiale Gestalt transformieren lassen, Proc. Imp. Acad. 6 2 (1930) 31-33 [euclid:pia/1195581421]

Textbook account:

• I. Martin Isaacs, theorem 5.12 in: Character theory of finite groups, Academic Press, New York (1976) [ISBN:978-0-8218-4229-4]

Further discussion:

• I. Martin Isaacs: Generalizations of Taketa’s Theorem on the Solvability of M-Groups, Proceedings of the American Mathematical Society 91 2 (1984) 192-194 [doi:10.2307/2044624, jstor:2044624]

• Yakov Berkovich: On the Taketa Theorem, Journal of Algebra 182 2 (1996) 501-510 [doi:10.1006/jabr.1996.0183]

• Tung Le, Jamshid Moori, Hung P. Tong-Viet: On a generalization of M-group, Journal of Algebra 374 (2013) 27-41 [arXiv:1206.5067, doi:10.1016/j.jalgebra.2012.10.018]