{'fraction_non_alphanumeric': 0.0723252971892012, 'fraction_numerical': 0.019442284190645483, 'mean_word_length': 3.384802727715538, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Monomial characters of finite solvable groups

We give new evidences to the fact that the structure of a solvable group can be controlled by irreducible monomial characters. In particular we inspect the role of monomial characters in Isaacs-Navarro-Wolf's conjecture and in Gluck's conjecture.

To conclude, we show that the characterization of the normality of a Sylow p-subgroup given in [MN12] can be checked by considering only monomial characters in the case of solvable groups.
Theorem C. Let G be a solvable groups, p a prime and consider P ∈ Syl p (G). Then P ⊴ G if and only if p does not divide χ(1), for every χ ∈ Irr m (G 1 P ).
The above results might lead to the following conceptual question for solvable groups: either monomial characters play an important role in the representation theory of solvable groups or they possess a large number of such characters. To support the first statement, consider G ∶= SL 2 (3) and define G n to be the direct product of n copies of G. Noticing that Irr m (G n ) Irr(G n ) = Irr m (G) Irr(G) n by [vdW88], we deduce that lim n→∞ Irr m (G n ) Irr(G n ) = 0 since G is a non-monomial group. This shows that there exist solvable groups with an arbitrarily small proportion of monomial characters.

Isaacs-Navarro-Wolf's conjecture
In [INW99], Isaacs-Navarro-Wolf's conjecture is proved for groups of odd order and, more generally, for solvable groups with abelian Sylow 2-subgroup. In [MW04] the authors show that N (G) ⊆ F 10 (G) for every solvable group and extend the previous result to every solvable group G in which every Fitting factor F i+1 (G) F i (G) has abelian Sylow 2-subgroup for every 1 ≤ i ≤ 9. Proof. Let u be the 2 ′ -part of x. We show that u ∈ F(G). By [DH92,13.8], it is enough to fix a non-Frattini chief factor K L of G and prove that u ∈ C G (K L) =∶ C. Since 1 ≠ F(G) ≤ C, induction on G yields u ∈ F G , where G ∶= G C. Furthermore G acts faithfully and irreducibly on Irr(K L).
Consider the quotient G L and observe that Lx ∈ N m (G L) and that K L satisfies the hypothesis of Lemma 2.1 in G L. It follows that Lx fixes a point in every (G L)-orbit on Irr(K L), and so that x fixes a point in every G-orbit on Irr(K L). In particular u fixes a point in each G-orbit on Irr(K L) and, by [INW99,Theorem 4.2], we conclude that u 2 = 1. This shows that u ∈ F(G) Next, we need analogues of [INW99,Theorem 4.4] and of [Yan09, Theorem 5.2] for monomial characters.
Proposition 2.3. Let G be a solvable group and assume there exists some x ∈ N m (G) ∖ F(G). Then there exists a subgroup F(G) ≤ N ⊴ G such that, if G ∶= G N , then x is an involution of F(G) and x ∉ A, for every abelian normal subgroup A ⊴ G.
Proof. Let M be a maximal element of the non-empty set 13.8], there exists a non-Frattini chief factor K L of G such that x ∉ C ∶= C G (K L). We claim that C is the normal subgroup we are looking for. Let G ∶= G C and observe that 1 ≠ x ∈ F(G) and that G acts faithfully and irreducibly on Irr(K L). By Lemma 2.1 we deduce that Lx fixes a point in each (G L)-orbit on Irr(K L). It follows that x fixes a point in each G-orbit on Irr(K L) and we conclude by [INW99, Theorem 4.2].
Proposition 2.4. Let G be a solvable group. Then there exists µ ∈ Irr m (F 8 (G)) such that µ G ∈ Irr m (G). In particular N m (G) ⊆ F 8 (G).
As a consequence we obtain the result mentioned at the beginning of the section.
The abelian group A acts faithfully on the completely reducible A-module Irr(F ). Noticing that, in this situation, complete reducibility coincide with the condition ( G , Irr(F ) ), we can find λ ∈ Irr(F ) such that A λ = 1 [INW99, Lemma 3.1]. In particular Z λ = F and ϑ ∶= λ Z ∈ Irr(Z). Since ϑ vanishes on Z ∖ F , and so do all its conjugates, we conclude that every character χ ∈ Irr(G ϑ) vanishes on Z ∖ F . On the other hand, observe that λ extends toλ ∈ Irr(G λ ) and that χ ∶=λ G ∈ Irr m (G). Since χ lies over ϑ, it follows that χ vanishes on Z ∖ F and so N m (G) ⊆ F .
Finally, assume that 4 does not divide the degree of any irreducible monomial character.
Lemma 2.7. Let G be a solvable group and suppose that p 2 does not divide χ(1), for all χ ∈ Irr m (G).
Then the Sylow p-subgroup of F i+1 (G) F i (G) has order 1 or p, for every i ≥ 1.
Proof. First, notice that it's enough to show the result for the Sylow , we may assume Φ(G) = 1. Now, G splits over the abelian normal subgroup F and every λ ∈ Irr(F ) has an extensionλ ∈ Irr(G λ ). By the Clifford correspondenceλ G ∈ Irr m (G) and so p 2 does not divide G ∶ G λ . In particular S ∶ S λ is either 1 or p. Next observe that, since Irr(F ) is a completely reducible and faithful (S F )-module, by [Wol99, Theorem A] there exists λ 1 ∈ Irr(F ) such that S ∶ F 1 2 ≤ S ∶ S λ 1 . In particular S F is abelian. This, together with [INW99, Lemma 3.1], implies that there exists λ 2 ∈ Irr(F ) such that F = S λ 2 and we conclude by the previous discussion.
Corollary 2.8. Conjecture A holds whenever G is a solvable group such that 4 does not divide χ(1), for all χ ∈ Irr m (G).
Proof. This follows by Corollary 2.5 and Lemma 2.7.
Notice that Lemma 2.7 is an adaptation of [LNW14, Lemma 2.1]. One may wonder if the main result of that paper holds by considering only monomial characters.
Problem 2.9. Let G be a solvable group and suppose that p 2 does not divide χ(1), for all χ ∈ Irr m (G). Is it true that G ∶ F(G) p ≤ p 2 ?
To end this section we mention that, by introducing the ideas used in this section in the setting of Brauer non-vanishing elements, one can prove a monomial version of the main result of [DPS17].
for every solvable group G. We show that Conjecture B holds in all the above mentioned situations. First, we collect some results on actions of solvable linear groups. For every integer n, we denote by π(n) the set of prime divisors of n.
Theorem 3.1. Let V be a finite faithful completely reducible G-module and π = π(V ). Define γ as follows: (ii) Set γ ∶= 1 2 if one of the following holds: • G is solvable and V ⋊ G has abelian Sylow 2-subgroup.
• G is solvable and V is primitive with either V ≠ 3 4 or V = 3 4 and G not conjugate to a subgroup of GL(V ) of order 1152.
Proof. Since, by hypothesis, F * (G Φ(G)) = F(G Φ(G)) = F(G) Φ(G), we may assume Φ(G) = 1. Now, F ∶= F(G) is abelian and V = Irr(F ) is a finite faithful completely reducible (G F )module. By Theorem 3.1 there exists λ ∈ V such that G λ ∶ F ≤ G ∶ F γ . Letλ ∈ Irr(G λ ) be an extension of λ and consider χ ∶=λ G ∈ Irr m (G). Then, We remark that Conjecture B holds also when V = 3 4 and G is conjugate to a subgroup of GL(V ) of order 1152. In this case GL(V ) has three conjugacy classes of subgroups of order 1152 and the subgroups of one of these classes do not have orbits of size at least 1152 1 2 . However, using GAP [GAP19] one can check that the result holds for this subgroups as well.
As we have already mentioned, there exist solvable groups in which b(G) is arbitrarily larger than b m (G).
Example 3.3. Let q and p be odd primes such that p ≡ −1 (mod q) and consider P an extraspecial p-group of order p 3 and exponent p. Let C ≤ Aut(P ) be a cyclic group of order q acting trivially on Z(P ) and define G ∶= P ⋊ C. By elementary character theory, we deduce that the only irreducible character degrees of G are 1, q and p and that no monomial characters can have degree p. This shows that b m (G) = q < p = b(G). Notice that the prime p can be chosen to be arbitrarily large by Dirichlet's theorem on arithmetic progressions.

A characterization of normal Sylow p-subgroups
In this final section, we consider the characterization given by Malle and Navarro in [MN12] and obtain Theorem C. We prove a slightly more general result which also implies [PL16, Theorem 1.3] and [PL17, Theorem 1.1 (2)].
Theorem 4.1. Let G be a finite group and set where χ runs over all characters in Irr m (G 1 P ) which vanish on some p-element of G. If G * m (p) is solvable, then it has a normal Sylow p-subgroup. Recall that G splits over M and letλ be an extension of λ to T ∶= G λ . Consider α 0 ∶=λ T ∩N and let α ∈ Irr(T M ) be the character corresponding to α 0 via the isomorphism T M ≃ T ∩ N . Then, β ∶=λα −1 is a linear character of T lying over λ and χ ∶= β G ∈ Irr m (G) by the Clifford correspondence. Notice that χ lies over 1 P by the MacKay formula. On the other hand, observe that χ lies over β, hence over λ and so over ϑ = λ K . By the previous paragraph, we deduce that L = G * m (p) ≤ Ker(χ). This implies that M ≤ Ker(ϑ) and so ϑ ∈ Irr(K M ). Since K M is abelian, we conclude that 1 = ϑ(1) = K ∶ M a contradiction.
As an immediate consequence, we obtain the following strong form of Theorem C.
Corollary 4.2. Let G be a solvable groups, p a prime and consider P ∈ Syl p (G). Then the following are equivalent: (i) P ⊴ G; (ii) p does not divide χ(1), for every χ ∈ Irr m (G 1 P ); (iii) χ(x) ≠ 0, for every χ ∈ Irr m (G 1 P ) and x ∈ P .
Proof. First notice that (i) implies (ii) and that (ii) implies (iii). Then we conclude by Theorem 4.1.