Some Remarks on Super $M_{p}$-groups

Let $G$ be a finite group and $p$ be a prime divisor of $|G|$. An irreducible $p$-Brauer character $φ$ of $G$ is called super-monomial if every primitive $p$-Brauer character inducing $φ$ is linear. The group $G$ is said to be a super $M_{p}$-group if every irreducible $p$-Brauer character of $G$ is super-monomial. In this note, we investigate the conditions under which a finite group $G$ qualifies as a super $M_{p}$-group. We demonstrate that every normal subgroup of a super $M_{p}$-group of odd order is an $M_{p}$-group.

💡 Research Summary

The paper introduces and studies a new class of finite groups called “super Mₚ‑groups”. Let G be a finite group and p a prime dividing |G|. An irreducible p‑Brauer character φ of G is called super‑monomial (or super‑monomial in the paper’s terminology) if every primitive p‑Brauer character that induces φ is linear. A group G is a super Mₚ‑group if every irreducible p‑Brauer character of G is super‑monomial. This definition parallels the classical notion of an M‑group (all complex irreducible characters are monomial) and the notion of an Mₚ‑group (all irreducible p‑Brauer characters are monomial). By construction, every super Mₚ‑group is an Mₚ‑group, and every super M‑group is an M‑group, but the converse implications do not hold in general.

The authors first recall the background on M‑groups, noting Taketa’s theorem that M‑groups are solvable, and the long‑standing question whether normal subgroups of M‑groups must also be M‑groups. Dade gave a counterexample, while Loukaki and Lewis proved that for groups of order pᵃqᵇ with p and q odd primes, normal subgroups inherit the M‑property. The paper then shifts to Brauer characters, citing Okuyama’s definition of monomial p‑Brauer characters and the fact that every M‑group is automatically an Mₚ‑group for any prime p (Fong–Swan theorem). However, the converse fails (e.g., GL(2, 3) is an M₂‑group but not an M‑group).

The central results are Theorem 1.1, Problem 1.2, Theorem 1.3, and Corollary 1.4.

Theorem 1.1 has two parts: (i) If every primitive p‑Brauer character of G is linear and every proper subgroup of G is an Mₚ‑group, then G is a super Mₚ‑group. The proof proceeds by considering an arbitrary irreducible p‑Brauer character φ. If φ is primitive, it is linear by hypothesis. If not, φ = λᴳ for some λ in a proper subgroup H; since H is an Mₚ‑group, λ is monomial, and consequently φ is super‑monomial. (ii) If G is a super M‑group, then G is a super Mₚ‑group for every prime p. The argument uses the relationship between a complex irreducible character χ and its restriction χ⁰ to p‑regular elements. Lemma 2.1 shows that if χ⁰ is primitive then χ is primitive. Since G is a super M‑group, any primitive χ is linear, which forces χ⁰ (hence any φ) to be super‑monomial.

Problem 1.2 asks whether every odd‑order Mₚ‑group is automatically a super Mₚ‑group. This is the Brauer‑character analogue of Isaacs’s conjecture that every odd‑order M‑group is a super M‑group. The authors note that a positive answer would imply Isaacs’s conjecture because when p does not divide |G|, p‑Brauer characters coincide with ordinary characters.

Theorem 1.3 shows that if the answer to Problem 1.2 is affirmative (i.e., every odd‑order Mₚ‑group is super Mₚ‑group), then every normal subgroup of an odd‑order Mₚ‑group is itself an Mₚ‑group. The proof relies on Lemma 2.4, which establishes that the class of Mₚ‑groups is closed under normal subgroups provided each member of the class is a super Mₚ‑group. Lemma 2.4 uses a combination of Clifford theory, the Mackey decomposition (Lemma 2.2), and the inheritance of monomiality for Brauer characters (Lemma 2.3). The argument proceeds by induction on the index of a normal subgroup, handling the two possibilities for the inertia group of an irreducible Brauer character (either it extends or induces irreducibly).

Corollary 1.4 observes that if Isaacs’s conjecture holds, then normal subgroups of odd‑order M‑groups are again M‑groups, reproducing a known result of Lewis.

The paper also includes several auxiliary lemmas:

• Lemma 2.1 connects primitivity of a Brauer character with primitivity of its ordinary lift.
• Lemma 2.2 (Mackey’s theorem) gives a double‑coset formula for induced Brauer characters.
• Lemma 2.3 shows that restriction of a monomial Brauer character to a subgroup remains monomial when irreducible.
• Lemma 2.4 establishes the closure under normal subgroups for the class of Mₚ‑groups assuming each is super Mₚ‑group.

Overall, the authors successfully extend the theory of monomial characters to the modular setting, define a natural “super” analogue, and prove that for groups of odd order the property of being a super Mₚ‑group forces normal subgroups to retain the Mₚ‑property. The results provide a clear pathway toward resolving Isaacs’s conjecture in the modular context and deepen the understanding of how Brauer character theory interacts with group structure, especially regarding normal subgroups and solvability.