Abstract
If p and q are primes, and G is a p-solvable finite group, it is possible to detect that a q-Sylow normalizer is contained in a p-Sylow normalizer using the character table of G. This is characterized in terms of the degrees of p-Brauer characters. Some consequences, which include yet another generalization of the Itô–Michler theorem, are also obtained.
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1 Introduction
If G is a finite group and \(\pi \) is a set of primes, let \(\mathrm{Irr}_{\pi }(G)\) be the subset of the irreducible complex characters \(\chi \) of G such that all the primes dividing the degree \(\chi (1)\) lie in \(\pi \). It is fair to say that the interaction between \(\mathrm{Irr}_{\pi }(G)\) and the structure of G is one of the recurrent problems in character theory. Recently, in [12], we have asked if it is possible to characterize group-theoretically when \(\mathrm{Irr}_{\pi }(G)=\mathrm{Irr}_{\rho }(G)\) for sets of primes \(\pi \) and \(\rho \).
In this paper, we fix a prime p, and we turn our attention to p-Brauer characters, within the universe of finite p-solvable groups. (Degrees of modular representations in characteristic p outside p-solvable groups are usually deemed an intractable subject.) If G is a p-solvable finite group and \(\mathrm{IBr}(G)\) is the set of the irreducible p-Brauer characters of G, we let \(\mathrm{IBr}_{\pi }(G)\) be the subset of \(\varphi \in \mathrm{IBr}(G)\) such that all the primes dividing \(\varphi (1)\) lie in \(\pi \). As usual, if r is a prime, \(r'\) denotes the set of primes different from r.
The following is our first main result.
Theorem A
Let G be a finite p-solvable group, and let q be a prime. Then, \(\mathrm{IBr}_{q'}(G) \subseteq \mathrm{IBr}_{p'}(G)\) if and only if there are \(Q \in \mathrm{Syl}_q(G)\) and \(P \in \mathrm{Syl}_p(G)\) such that \(\mathbf{N}_{G}(Q) \subseteq \mathbf{N}_{G}(P)\).
Theorem A was the main result of [1], assuming that G is both p-solvable and q-solvable. To better understand our this new situation, we invite the reader to consider, for instance, \(G=\mathrm{PSL}_2(3^5).5\), for \(p=5\). If \(q=2\), then \(\mathbf{N}_{G}(P)=\mathbf{N}_{G}(Q)\) for some \(p \in \mathrm{Syl}_p(G)\) and \(Q\in \mathrm{Syl}_q(G)\). Of course, G is p-solvable, but not q-solvable. There are families of almost-simple groups like this, and we are able to deal with this new situation thanks to the main result of [6]. In particular, the proof of Theorem A depends on the Classification of Finite Simple Groups.
From Theorem A, and using McKay bijections for Brauer characters in p-solvable groups together with the recent proof of the divisibility of degrees between Glauberman correspondents by M. Geck in [2], we can prove our second main result.
Theorem B
Let G be a finite p-solvable group, and let q be a prime different from p. Then \(\mathrm{IBr}_{p'}(G)=\mathrm{IBr}_{q'}(G)\) if and only if there are \(P \in \mathrm{Syl}_p(G)\) and \(Q \in \mathrm{Syl}_q(G)\) such that \(\mathbf{N}_{G}(P)=P\mathbf{N}_{G}(Q)\) and Q is abelian.
As the reader can easily check, in the trivial case where p does not divide |G|, Theorem B is yet another restatement of the Itô–Michler theorem.
As happens with complex irreducible characters (see [12]), it does not seem easy to group-theoretically characterize when \(\mathrm{IBr}_\pi (G)=\mathrm{IBr}_\rho (G)\) for arbitrary sets of primes \(\pi \) and \(\rho \), even if G is p-solvable. In the case, for instance, where \(\pi =\mathbb P\) is the set of all primes and \(\rho =q'\), where q is a prime, this constitutes Problem 3.2 of [9]. (This problem was studied long before in [8] and more recently in [5].) It has now been conjectured that \(\mathrm{IBr}_{q'}(G)=\mathrm{IBr}_{\mathbb P}(G)\) if and only if the number of p-regular classes of G is the number of p-regular classes of \(\mathbf{N}_{G}(Q)/Q'\), where \(Q \in \mathrm{Syl}_q(G)\). This is a consequence of the Inductive McKay conjecture, and it seems difficult to obtain a direct proof. (See Conjecture D in [7].)
2 Proof of theorem A
Our notation for Brauer characters follows [9]. The deepest part of the proof of Theorem A comes from the main result in [MN].
Theorem 2.1
Let G be a finite \(\pi \)-separable group. Let H be a Hall \(\pi \)-subgroup, let K be a \(\pi \)-complement of G, and let q be a prime. Then, every \(\alpha \in \mathrm{Irr}_{q'}(H)\) extends to G if and only if there is \(Q \in \mathrm{Syl}_q(H)\) such that \(\mathbf{N}_{G}(Q) \subseteq \mathbf{N}_{G}(K)\).
Proof
This is Theorem A of [6]. \(\square\)
A very useful result to deal with the hypotheses in this paper appears in Suzuki’s book.
Theorem 2.2
Let G be a p-solvable group, and let q be a prime. Let \(\pi =\{p,q\}\). Then, G has a unique conjugacy class of Hall \(\pi \)-subgroups, and every \(\pi \)-subgroup of G is contained in one of them.
Proof
This follows from 5.3.13 of [14]. \(\square\)
For the reader’s convenience, let us prove the following standard result.
Lemma 2.3
Let G be a finite group. Let Q be a Sylow q-subgroup of G and let N be a normal subgroup of G. If \(\varphi \in \mathrm{IBr}_{q'}(G)\), then \(\varphi _N \) has a Q-invariant irreducible constituent and any two of them are \(\mathbf{N}_{G}(Q)\)-conjugate.
Proof
Let \(\varphi \in \mathrm{IBr}_{q'}(G)\) and let \(\theta \in \mathrm{IBr}(N)\) be an irreducible constituent of \(\varphi _N\). Let \(G_\theta \) be the stabilizer of \(\theta \) in G. By Clifford’s theorem (see Corollary 8.9 of [9], for instance), we have that \(|G:G_\theta |\) divides \(\varphi (1)\), and hence, it is a \(q'\)-number. It follows that there is an element \(g\in G\) such that \(Q^g\) is contained in \(G_\theta \). Hence, \(Q\subseteq G_\theta ^{g^{-1}}=G_{\theta ^{g^{-1}}}\) and \(\theta ^{g^{-1}}\) is a Q-invariant irreducible constituent of \(\varphi _N\).
Now, suppose that \(\theta \) and \(\mu \) are two Q-invariant irreducible constituents of \(\varphi _N\). Again by Clifford’s theorem we have that there is an element \(g\in G\) such that \(\mu =\theta ^g\). It follows that Q and \(Q^g\) are contained in \(G_\mu \), and hence, there is an element \(x\in G_\mu \) such that \(Q=(Q^g)^x\). Therefore \(gx\in \mathbf{N}_{G}(Q)\) and \(\theta ^{gx}=(\theta ^g)^x=\mu ^x=\mu \), as wanted. \(\square\)
The following result implies Theorem A. In its proof, we shall use Fong characters of Brauer characters, and we refer the reader to Chapter 10 of [9] for their main properties. (The term Fong character was coined by I. M. Isaacs after some results of P. Fong.) We also use the fact that if G is p-solvable, H is a p-complement of G, and \(\varphi \in \mathrm{IBr}(G)\) has degree not divisible by p, then \(\varphi _H \in \mathrm{Irr}(H)\). (See Theorem 10.9 of [9].) A complication when dealing with Brauer characters is that we do not have any form of Frobenius reciprocity, even in favorable conditions. For instance, in the previous situation where H is a p-complement of a p-solvable group G, if \(\alpha \in \mathrm{Irr}(H)\), and \(\varphi \) is an irreducible constituent of \(p'\)-degree of the induced Brauer character \(\alpha ^G\), then \(\alpha \) needs not be the irreducible character \(\varphi _H\); a fact that would simplify our proof below. (If \(G=\mathsf{A}_4\), \(p=2\), and \(H=\mathsf{C}_3\), then the Brauer character \((1_H)^G=21_G + \lambda _1 + \lambda _2\), where \(\lambda _i\) are distinct linear Brauer characters. Now take \(\alpha =1_H\) and \(\varphi =\lambda _i\).)
Theorem 2.4
Let G be a p-solvable group, let \(\mathrm{IBr}(G)\) be the set of irreducible Brauer characters of G, and let \(q \ne p\) be a prime. Let \(P \in \mathrm{Syl}_p(G)\) and \(Q \in \mathrm{Syl}_q(G)\) such that \(U=PQ\) is a subgroup of G. Suppose that H is a p-complement of G containing Q. Then, the following are equivalent.
-
(a)
\(\mathrm{IBr}_{q'}(G) \subseteq \mathrm{IBr}_{p'}(G)\).
-
(b)
\(\mathbf{N}_{G}(Q) \subseteq \mathbf{N}_{G}(P)\).
-
(c)
Every \(\alpha \in \mathrm{Irr}_{q'}(H)\) extends to G.
Proof
Set \(\pi =\{p,q\}\). By Theorem 2.2, notice that we can find \(P \in \mathrm{Syl}_p(G)\) and \(Q \in \mathrm{Syl}_q(G)\) such that \(U=PQ\) is a Hall \(\pi \)-subgroup of G. (In fact, given a Hall \(\pi \)-subgroup U of G, then \(U=PQ\) for every \(P \in \mathrm{Syl}_p(U)\) and \(Q\in \mathrm{Syl}_q(U)\).)
Assume (a). We prove (b) by induction on |G|. If K is a minimal normal subgroup of G, then we have that \(\mathbf{N}_{G}(Q) \subseteq \mathbf{N}_{G}(P) K\), by using induction in G/K. Since G is p-solvable, then K is either a p-group or a \(p'\)-group. In the first case, \(\mathbf{N}_{G}(P) K=\mathbf{N}_{G}(P)\) and we are done. So we assume that K is a \(p'\)-group.
Let V be any subgroup of G containing KQ and let \(\delta \in \mathrm{IBr}_{q'}(V)\). By Lemma 2.3, there exists a Q-invariant irreducible constituent \(\tau \in \mathrm{IBr}(K)\) of the restriction \(\delta _K\) (using that \(Q \in \mathrm{Syl}_q(V)\)). We claim that \(\tau \) is also P-invariant. By Corollary 8.7 of [9], we have that \(\delta \) is an irreducible constituent of the induced Brauer character \(\tau ^V\). Now consider the Brauer character \(\delta ^G\), which has degree \(|G:V|\delta (1)\), which is not divisible by q. Hence, there exists an irreducible constituent \(\varphi \in \mathrm{IBr}(G)\) of \(\delta ^G\) of degree not divisible by q. By hypothesis, we have that \(\varphi \) has degree not divisible by p. Now, \(\varphi \) is an irreducible constituent of \(\tau ^G\), and therefore, \(\tau \) is an irreducible constituent of the restriction \(\varphi _K\), again by Corollary 8.7 of [9]. If \(I=G_\tau \) is the stabilizer of \(\tau \) in G, and \(\mu \in \mathrm{IBr}(I)\) is the Clifford correspondent of \(\varphi \) over \(\tau \), we have that |G : I| is a \(\pi '\)-number. Therefore, using Theorem 2.2, we have that \(U \subseteq I^g\) for some \(g \in G\). Then, \(\tau \) and \(\tau ^g\) are Q-invariant constituents of \(\varphi \), and thus, by Lemma 2.3, we have that \(\tau ^g=\tau ^x\), for some \(x \in \mathbf{N}_{G}(Q) \subseteq K\mathbf{N}_{G}(P)\). Hence, we may assume that \(\tau ^g=\tau ^y\) for some \(y \in \mathbf{N}_{G}(P)\). Then, \(I^{gy^{-1}}=I\), \(U^{y^{-1}} \subseteq I\), and we conclude that \(P=P^{y^{-1}} \subseteq I\). In other words, \(\tau \) is P-invariant, as claimed.
Now, let \(V=\mathbf{N}_{G}(P) K\). We claim that \(\mathrm{IBr}_{q'}(V) \subseteq \mathrm{IBr}_{p'}(V)\). Let \(\delta \in \mathrm{IBr}_{q'}(V)\). By the claim in the previous paragraph, let \(\tau \in \mathrm{IBr}(K)\) be PQ-invariant under \(\delta \). By Theorem 8.11 of [9], there is a unique \(\hat{\tau }\in \mathrm{IBr}(KP)\) over \(\tau \), extending \(\tau \), and we conclude that \(\delta \) lies over \(\hat{\tau }\). Now, \(KP \triangleleft \,V\), V/KP is a \(p'\)-group, and we have that \(\delta (1)/\hat{\tau }(1)\) divides |V : KP| by Theorem 8.30 of [9]. Since \(\tau (1)\) is not divisible by p, we conclude that \(\delta \) has \(p'\)-degree, as claimed. By induction, we may assume that \(V=G\). That is, \(KP \triangleleft \,G\). We want to use Theorem 2.1.
Next, we show that every \(\alpha \in \mathrm{Irr}_{q'}(H)\) extends to G. By Lemma 2.3, let \(\tau \in \mathrm{Irr}(K)\) be Q-invariant under \(\alpha \). By the claim in the third paragraph, we have that \(\tau \) is P-invariant too. By Corollary 6.28 of [3], we have that \(\tau \) has a canonical extension \(\gamma \in \mathrm{Irr}(KP)\). Using the uniqueness of \(\gamma \), we easily check that \(G_\gamma \cap H=H_\tau \). By Isaacs restriction theorem (Lemma 6.8(d) of [11]), the Clifford correspondence and Mackey’s theorem (Theorem 1.16 of [10]), we have that \(\alpha \) extends to G. If \(\pi _0=p'\) is the set of primes dividing |G| different from p, by Theorem 2.1 applied to \(\pi _0\), we conclude that there is \(Q_1 \in \mathrm{Syl}_q(H)\) such that \(\mathbf{N}_{G}(Q_1) \subseteq \mathbf{N}_{G}(P)\). In particular, \(P \triangleleft \,PQ_1\). By Theorem 2.2, \(PQ_1\) and PQ are G-conjugate. Hence \(P\triangleleft \,PQ\) and therefore \(Q, Q_1 \in \mathrm{Syl}_q(\mathbf{N}_{G}(P))\). Hence, \(Q_1^z=Q\) for some \(z\in \mathbf{N}_{G}(P)\), and \(\mathbf{N}_{G}(Q)=\mathbf{N}_{G}(Q_1)^z \subseteq \mathbf{N}_{G}(P)\).
We have that (b) implies (c), by Theorem 2.1 applied to \(\pi _0=p'\).
Finally, we prove that (c) implies (a) by using Fong characters. Suppose now that every \(\alpha \in \mathrm{Irr}_{q'}(H)\) extends to G. We show that \(\mathrm{IBr}_{q'}(G) \subseteq \mathrm{IBr}_{p'}(G)\). Let \(\varphi \in \mathrm{IBr}_{q'}(G)\), and let \(\alpha \in \mathrm{Irr}(H)\) be an irreducible constituent of \(\varphi _H\) such that \(\alpha (1)=\varphi (1)_{p'}\) (using Theorem 10.18 of [9].) In other words, \(\alpha \) is a Fong character for \(\varphi \). Then, \(\alpha (1)\) is not divisible by q, and by hypothesis, we have that \(\alpha \) extends to some \(\chi \in \mathrm{Irr}(G)\). Then, the Brauer character \(\mu =\chi ^0 \in \mathrm{IBr}(G)\) extends \(\alpha \). (Indeed, if \(\chi ^0=\varphi _1+\varphi _2\) for Brauer characters \(\varphi _i\) of G, then the irreducible character \(\alpha \) would be written as \((\varphi _1)_H+(\varphi _2)_H\).) By Theorem 10.17 of [9], we have that \(\varphi =\mu \), and \(\varphi (1)=\mu (1)=\alpha (1)\) has degree not divisible by p. \(\square\)
3 Proof of theorem B
If N is a normal subgroup of G and \(\theta \in \mathrm{IBr}(N)\), we write \(\mathrm{IBr}(G\,|\,\theta )\) to denote the set of irreducible Brauer characters \(\varphi \) of G such that \(\theta \) is an irreducible constituent of \(\varphi _N\). We write \(\mathrm{IBr}_{p'}(G\,|\,\theta )=\mathrm{IBr}_{p'}(G)\cap \mathrm{IBr}(G\, |\, \theta )\).
Next is the version for Brauer characters and p-solvable groups of Theorem A of [13], which we shall need to prove Theorem B. It is worth mentioning that its proof uses the recent proof of the divisibility of the degrees of the Glauberman correspondence in [2].
Theorem 3.1
Let G be a p-solvable group, and \(P\in \mathrm{Syl}_p(G)\), then there is a bijection \(f: \mathrm{IBr}_{p'}(G)\rightarrow \mathrm{IBr}_{p'}(\mathbf{N}_{G}(P))\) such that \(f(\varphi )(1)\) divides \(\varphi (1)\) for all \(\varphi \in \mathrm{IBr}_{p'}(G)\). Furthermore, \(\varphi (1)/f(\varphi ) (1)\) divides \(|G:\mathbf{N}_{G}(P)|\).
Proof
We argue by induction on |G|. Since \(\mathbf{O}_{p}(G)\) is in the kernel of every \(\varphi \in \mathrm{IBr}(G)\), by induction we may assume that \(\mathbf{O}_{p}(G) =1\) and hence \(K=\mathbf{O}_{p'}(G)>1\). Let \(S/K=\mathbf{O}_{p}(G/K)\) and notice that \(P_0=P\cap S\) is a Sylow p-subgroup of S. By the Frattini argument, we have that \(G=K\mathbf{N}_{G}(P_0)\). Notice also that \(\mathbf{N}_{G}( P_0)<G\) since \(\mathbf{O}_{p}(G) =1\).
Let \(\theta _1, \ldots , \theta _s\) be a complete set of representatives of the orbits of the action of \(\mathbf{N}_{G}(P)\) on the P-invariant irreducible characters of K. By Lemma 2.3, we have that
is a disjoint union. Fix \(\theta _i \in \mathrm{Irr}(K)\), P-invariant, and observe that \(\theta _i\) is also \(P_0\)-invariant. Let \(\theta _i^*\in \mathrm{Irr}(\mathbf{C}_{K}(P_0))= \mathrm{IBr}(\mathbf{C}_{K}(P_0))\) be the Glauberman correspondent of \(\theta _i\) and let \(T_i=G_{\theta _i}\) be the stabilizer of \(\theta _i\) in G. Since the Glauberman correspondence and the action of \(\mathbf{N}_{G}(P_0)\) commute (see Lemma 2.10 of [10]), it follows that \( \mathbf{N}_{T_i}(P_0) =T_i\cap \mathbf{N}_{G}(P_0)\) is the stabilizer of \(\theta _i^*\) in \(\mathbf{N}_{G}(P_0)\). By Dade’s theorem (see Theorem (6.5) of [15]), we have that \((T_i, K, \theta _i)\) and \( (\mathbf{N}_{T_i}(P_0), \mathbf{C}_{K}(P_0), \theta _i^*)\) are isomorphic character triples. In particular, there is a bijection \(\Delta :\mathrm{Irr}(T_i|\theta _i)\rightarrow \mathrm{Irr}(\mathbf{N}_{T_i}(P_0)|\theta _i^*)\) such that \(\chi (1)/\theta _i(1)=\Delta (\chi )(1)/\theta _i^*(1)\) for all \(\chi \in \mathrm{Irr}(T_i|\theta _i)\). Now using \(\Delta \) and Lemma 3.12 of [4] we can construct a bijection \( ^*: \mathrm{IBr}(T_i\, |\, \theta _i) \rightarrow \mathrm{IBr}( \mathbf{N}_{T_i}(P_0) \, |\, \theta _i^*)\) such that \( \psi (1)/\theta _i(1)= \psi ^*(1)/\theta _i^*(1)\) for all \(\psi \in \mathrm{IBr}(T_i\, |\, \theta _i) \), and since \(\theta _i(1)\) and \(\theta _i^*(1)\) are \(p'\)-numbers we have that \( ^*: \mathrm{IBr}_{p'}({T_i\, |\, \theta _i}) \rightarrow \mathrm{IBr}_{p'}( { \mathbf{N}_{T_i}(P_0) \, |\, \theta _i^*})\) is a bijection.
Let \(\chi \in \mathrm{IBr}_{p'}(G)\) and let \(\theta _i\in \mathrm{IBr}(K) =\mathrm{Irr}(K)\) be such that \( \chi \in \mathrm{IBr}_{p'}( {G\,|\,\theta _i})\). By the Clifford correspondence (Theorem 8.9 of [9]), we have that there is \( \psi \in \mathrm{IBr}_{p'}(T_i\, |\, \theta _i)\) such that \(\psi ^G=\chi \). Since \(\mathbf{N}_{T_i}(P_0)\) is the stabilizer of \(\theta _i^*\) in \(\mathbf{N}_{G}(P_0)\), again by the Clifford correspondence we have that \((\psi ^*)^{\mathbf{N}_{G}(P_0)}\in \mathrm{Irr}(\mathbf{N}_{G}(P_0))\). Furthermore, since \(P\subseteq \mathbf{N}_{T_i}(P_0)\), we have that
is a \(p'\)-number. Hence, we define
by \(g(\chi )=(\psi ^*)^{\mathbf{N}_{G}(P_0)}\).
Since \(\theta _1, \ldots , \theta _s\) is a complete set of representatives of the action of \(\mathbf{N}_{G}(P)\) on the P-invariant characters of K and the Glauberman correspondence commutes with the action of \(\mathbf{N}_{G}(P)\subseteq \mathbf{N}_{G}(P_0)\), we have that \(\theta _1^*, \ldots , \theta _s^*\) is a complete set of representatives of the action of \(\mathbf{N}_{G}(P)\) on the P-invariant irreducible characters of \(\mathrm{Irr}(\mathbf{C}_{K}(P_0))=\mathrm{IBr}(\mathbf{C}_{K}(P_0))\). By Lemma 2.3, we have that
is a disjoint union and it follows that g is a bijection.
Let \(\chi \in \mathrm{IBr}_{p'}(G|\theta _i)\) and let \(\psi \in \mathrm{IBr}_{p'}(T_i|\theta _i)\) with \(\psi ^G=\chi \). Then \(\psi (1) = \psi ^*(1)\theta _i(1)/\theta ^*_i(1)\) and, since \(\theta ^*_i(1)\) divides \(\theta _i(1)\) (by the recent main theorem of [2]), we have that \(\psi ^*(1)\) divides \(\psi (1)\). Since \(G=\mathbf{N}_{G}(P_0) K \) we have that \(g(\chi )(1)= |\mathbf{N}_{G}(P_0) :\mathbf{N}_{T_i}(P_0) |\psi ^*(1)\) divides \(|G:T_i|\psi (1)=\chi (1)\).
Finally, since \(\mathbf{N}_{G}(P_0)<G\), we apply induction to obtain a bijection
such that \(h(\psi )(1)\) divides \(\psi (1)\) and \(\chi (1)/h(\chi )(1)\) divides \(|\mathbf{N}_{G}(P_0): \mathbf{N}_{G}(P)|\) for all \(\psi \in \mathrm{IBr}_{p'}(\mathbf{N}_{G}(P_0))\). Let \(f=gh=h\circ g\). Clearly, f is a bijection and \(f(\chi )(1)\) divides \(\chi (1)\) for all \(\chi \in \mathrm{IBr}_{p'}(G)\). Now, since \( g(\chi )(1)/h(g(\chi ))(1)\) divides \(|\mathbf{N}_{G}(P_0): \mathbf{N}_{G}(P)|\), we have that \(\chi (1)/f(\chi )(1)\) divides \(|\mathbf{N}_{G}(P_0): \mathbf{N}_{G}(P)|\psi (1)/\psi ^*(1)=|\mathbf{N}_{G}(P_0): \mathbf{N}_{G}(P)|\theta _i(1)/\theta _i^*(1)\). By Problem 13.2 of [3], \(\theta _i(1)/\theta _i^*(1)\) divides \(|K:\mathbf{C}_{K}(P_0)|=|G:\mathbf{N}_{G}(P_0)|\). Hence, \(\chi (1)/f(\chi )(1)\) divides \(|G:\mathbf{N}_{G}(P)|\). \(\square\)
The following is Theorem B.
Theorem 3.2
Suppose that G is a p-solvable finite group and let q be a prime different from p. Then
if and only if there is a Sylow p-subgroup P of G and a Sylow q-subgroup Q of G, such that \(\mathbf{N}_{G}(P) =P\mathbf{N}_{G}(Q)\) and Q is abelian.
Proof
Suppose that \(\mathrm{IBr}_{q'}(G)=\mathrm{IBr}_{p'}(G)\). By Theorem 2.4, we have that there is a Sylow p-subgroup of G and a Sylow q-subgroup of G such that \(\mathbf{N}_{G}(Q)\subseteq \mathbf{N}_{G}(P)\). We claim that
First, we notice that Theorem 2.4 applied to \(\mathbf{N}_{G}(P)\) shows \(\mathrm{IBr}_{q'}(\mathbf{N}_{G}(P) )\subseteq \mathrm{IBr}_{p'}(\mathbf{N}_{G}(P))\). Let \(\mu \in \mathrm{IBr}_{p'}(\mathbf{N}_{G}(P))\). By Theorem 3.1, there is \(\varphi \in \mathrm{IBr}_{p'}(G)\) such that \(\mu (1)\) divides \(\varphi (1)\) which is a \(q'\)-number. Hence, we conclude \(\mu \in \mathrm{IBr}_{q'}(\mathbf{N}_{G}(P))\). Then \(\mathrm{IBr}_{q'}(\mathbf{N}_{G}(P) )= \mathrm{IBr}_{p'}(\mathbf{N}_{G}(P))\), as claimed.
Therefore, we may assume, arguing by induction that \(P\triangleleft \,G\). Then, we have that \(\mathrm{IBr}(G)=\mathrm{IBr}(G/P)=\mathrm{Irr}(G/P)\) and
By the Itô-Michler theorem, we know that G/P has a normal and abelian Sylow q-subgroup PQ/Q. Hence, Q is abelian and \(PQ\triangleleft \,G\). Then \(G=P\mathbf{N}_{G}(Q)\) by the Frattini’s argument.
Conversely, suppose that there is \(P\in \mathrm{Syl}_p(G)\) and \(Q\in \mathrm{Syl}_q(G)\) such that \(\mathbf{N}_{G}(P) =P\mathbf{N}_{G}(Q)\) and Q is abelian. Notice that since \(\mathbf{N}_{G}(Q) \subseteq \mathbf{N}_{G}(P)\) by Theorem 2.4 we have that \(\mathrm{IBr}_{q'}(G)\subseteq \mathrm{IBr}_{p'}(G)\). We only need to prove the reverse containment. If \(\mathbf{N}_{G}(P)<G\), arguing by induction, we have that \(\mathrm{IBr}_{q'}(\mathbf{N}_{G}(P) )= \mathrm{IBr}_{p'}(\mathbf{N}_{G}(P))\). Let \(\varphi \in \mathrm{IBr}_{p'}(G)\) and let \(f:\mathrm{IBr}_{p'}(G)\rightarrow \mathrm{IBr}_{p'}(\mathbf{N}_{G}(P))\) be the bijection given in Theorem 3.1. Hence \(\varphi (1)/ f(\varphi )(1)\) divides \(|G:\mathbf{N}_{G}(P)|\) which is not divisible by q. Since \(\mathrm{IBr}_{q'}(\mathbf{N}_{G}(P) )= \mathrm{IBr}_{p'}(\mathbf{N}_{G}(P))\), we have that \(f(\varphi )(1)\) is not divisible by q and thus \(\varphi \in \mathrm{IBr}_{q'}(G)\).
Hence, we may assume that P is a normal subgroup of G and then \(\mathrm{IBr}(G)=\mathrm{Irr}(G/P)\). Also \(G=\mathbf{N}_{G}(Q) P\) and hence PQ is normal in G. Then, PQ/P is an abelian normal Sylow q-subgroup of G/P. By Itô’s theorem, we have that \(\mathrm{Irr}(G/P)=\mathrm{Irr}_{q'}(G/P)\). It follows that
and we are done. \(\square\)
To prove the assertion in the Abstract, it is enough to notice that in p-solvable groups, the ordinary character table of G uniquely determines the Brauer characters of G, by using the Fong–Swan theorem. (See Theorem 10.1 and Corollary 10.4 of [9].)
Finally, using Isaacs \(\pi \)-characters, the Glauberman–Isaacs correspondence, and some ad hoc arguments, it is possible to replace in Theorems A and B of this paper \(p'\) by \(\pi \), p-solvable groups by \(\pi \)-separable groups, and Sylow p-subgroups by Hall \(\pi \)-complements.
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This research is supported by Ministerio de Ciencia e Innovación PID2019-103854GB-I00 and FEDER funds. The third and fourth authors are also supported by Generalitat Valenciana AICO/2020/298. The third author also acknowledges support by Ministerio de Ciencia e Innovación PID2020-118193GA-I00 and “Convocatoria de contratación para la especialización de personal investigador doctor en la UPV/EHU (2019)”. The second author thanks G. Malle for the example after Theorem A. This work was done, while the first author visited the University of Valencia.
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Bonazzi, L., Navarro, G., Rizo, N. et al. Brauer character degrees and Sylow normalizers. Annali di Matematica 201, 2575–2582 (2022). https://doi.org/10.1007/s10231-022-01210-0
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DOI: https://doi.org/10.1007/s10231-022-01210-0