Finite Solvable Groups Research Articles

Taketa’s theorem for some character degree sets

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then $${{\rm dl}(G) \leqslant |{\rm cd} (G)|}$$ , where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality $${{\rm dl}(N) \leqslant |{\rm cd} {(G \mid N)}|}$$ holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if $${{\rm cd} {(G \mid N')}}$$ is a set of non-trivial p–powers for some fixed prime p.

Injectors with a central socle in a finite solvable group

In response to an Open Question of Doerk and Hawkes (1992) [2, IX §4, p. 628], we shall describe three constructions for the Zπ-injectors of a finite solvable group, where Zπ is the Fitting class formed by the finite solvable groups whose π-socle is central (and π is a set of prime numbers).

On the Northcott property and other properties related to polynomial mappings

AbstractWe prove that if K/ℚ is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/ℚ has the Northcott property.Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km/ℚ)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in ℚ(d).We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.

Finite solvable groups with all n-maximal subgroups $$\mathfrak{U}$$ -subnormal

We describe the finite solvable groups with all n-maximal subgroups \(\mathfrak{U}\)-subnormal.

Groups where each element is conjugate to its certain power

AbstractThis paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power g n. Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.

A result related to non-vanishing elements of finite solvable groups

A result related to non-vanishing elements of finite solvable groups

On finite groups with their character tables having at most "Equation missing" zeros in each column

Abstract In the present paper, the authors classify finite solvable groups whose character tables have at most p + 1 zeros in each column, where p is the minimal prime divisor of their orders. MSC:20C15.

On non-Frattini chief factors and solvability of finite groups

A subgroup H of a group G is said to be a semi CAP*-subgroup of G if there is a chief series 1 = G0 < G1 < ⋯ < Gm = G of G such that for every non-Frattini chief factor Gi/Gi − 1, H either covers Gi/Gi − 1 or avoids Gi/Gi − 1. In this paper, some sufficient conditions for a normal subgroup of a finite group to be solvable are given based on the assumption that some maximal subgroups are semi CAP*-subgroups.

Finite solvable groups in which semi-normality is a transitive relation

A subgroup H of a finite group G is said to be seminormal in G if every Sylow p-subgroup of G, p a prime, with (|H|, p) = 1 normalizes H. A group G is called an SNT-group if seminormality is a transitive relation in G. Properties of solvable SNT-groups are studied. For example, subgroups of solvable SNT-groups are SNT-groups.

Chains of modular elements and shellability

Let L be a lattice admitting a left-modular chain of length r, not necessarily maximal. We show that if either L is graded or the chain is modular, then the (r−2)-skeleton of L is vertex-decomposable (hence shellable). This proves a conjecture of Hersh. Under certain circumstances, we can find shellings of higher skeleta. For instance, if the left-modular chain consists of every other element of some maximum length chain, then L itself is shellable. We apply these results to give a new characterization of finite solvable groups in terms of the topology of subgroup lattices.Our main tool relaxes the conditions for an EL-labeling, allowing multiple ascending chains as long as they are lexicographically before non-ascending chains. We extend results from the theory of EL-shellable posets to such labelings. The shellability of certain skeleta is one such result. Another is that a poset with such a labeling is homotopy equivalent (by discrete Morse theory) to a cell complex with cells in correspondence to weakly descending chains.

Some new characterizations of solvable PST-groups

A subgroup H of a finite group G is said to be τ-semipermutable in G if it permutes with all the Sylow subgroups Q of G such that (|H|, |Q|) = 1 and (|H|, |QG|) ≠ 1. A rather remarkable result of Lukyanenko and Skiba (Rend Semin Mat Univ Padova, 124:231–246, 2010) is: a finite solvable group G is a PST-group if and only if every subgroup of Fit(G) is τ-semipermutable. A local version of this result is established in this paper. A subgroup H of G is said to be τ-seminormal provided that it is normalized by all Sylow subgroups Q such that (|H|, |Q|) = 1 and (|H|, |QG|) ≠ 1. It is shown that a finite solvable group is a PST-group if and only if every subgroup of Fit(G) is τ-seminormal in G.

A criterion for subnormality of subgroups in finite groups

Based on Wielandt’s criterion for subnormality of subgroups in finite groups, we study 2-maximal subgroups of finite groups and present another subnormality criterion in finite solvable groups.

Regular and p-regular orbits of solvable linear groups

Let V be a faithful G-module for a finite group G and let p be a prime dividing |G|. An orbit vG for the action of G on V is p-regular if |vG|p=|G:CG(v)|p=|G|p. Zhang asks the following question in Zhang (1993) [8]. Assume that a finite solvable group G acts faithfully and irreducibly on a vector space V over a finite field F. If G has a p-regular orbit for every prime p dividing |G|, is it true that G will have a regular orbit on V? In Lü and Cao (2000) [4], Lü and Cao construct an example showing that the answer to this question is no, however the example itself is not correct. In this paper, we study Zhangʼs question in detail. We construct examples showing that the answer to this question is no in general. We also prove the following result. Assume a finite solvable group G of odd order acts faithfully and irreducibly on a vector space V over a field of odd characteristic. If G has a p-regular orbit for every prime p dividing |G|, then G will have a regular orbit on V.

Navarro vertices and normal subgroups in groups of odd order

Let p be a prime and suppose G is a finite solvable group and χ is an ordinary irreducible character of G. Navarro has shown that one can associate to χ a pair (Q, δ), where Q is a p-subgroup of G and δ is an irreducible character of Q, which is unique up to conjugacy. This pair is called the Navarro vertex of χ, and in this paper we study the behavior of the Navarro vertices of χ with respect to normal subgroups of G when G is assumed to have odd order and χ is a lift of an irreducible Brauer character of G. For NCG, we use these results to give a sufficient condition for the constituents of χN to be lifts of Brauer characters, and we develop a Clifford-type correspondence for lifts with a given Navarro

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C G (v) ⊆ F 2(G) (where F 2(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C H (v) ⊆ O π(G).

On the F-hypercenter of finite solvable groups

Let F be a non-empty subgroup-closed saturated formation, it is studied that the structure of a finite group G under the assumption that every subgroup of prime order p if p≠ 2 and every cyclic subgroup of order 2 or 4 if p=2 are contained in the F-hypercenter of the normalizer N G (P) of a Sylow p-subgroup P of G. Our results improve and extend some conclusions in the previous references.

Character degree graphs with no complete vertices

Let Γ be a graph in which each vertex is non-adjacent to another different one. We show that, if G is a finite solvable group with abelian Fitting subgroup and with character degree graph Γ ( G ) = Γ , then G is a direct product of subgroups having a disconnected character degree graph. In particular, Γ is a join of disconnected graphs. We deduce also that solvable groups with abelian Fitting subgroup have a character degree graph with diameter at most 2.

Some Character Degree Conditions Implying Solvability of Finite Groups

Let cd(G) be the set of all irreducible complex character degrees of a finite group G. In this paper, we show that if a, b, and c are pairwise relatively prime integers and \({\rm{cd}} \left(G\right) \subseteq \{1, a, b, c, ab, ac\}\), then either G is solvable or {a, b, c} = {2 f − 1, 2 f , 2 f + 1} for some \(f \geqslant 2\) and cd (G) = {1, a, b, c}.

Large orbits of odd-order subgroups of solvable linear groups

Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be an odd-order subgroup of G, then H has at least two regular orbits on V ⊕ V . Suppose in addition that | V | is odd, then there exists v ∈ V in a regular orbit of F ( G ) ∩ H such that C H ( v ) ⊆ F 2 ( G ) . Let G be a solvable group, H be an odd-order Hall π-subgroup of G, V be a faithful G-module, over possibly different finite fields of odd π-characteristic and assume that V O π ( G ) is completely reducible, then there exists v ∈ V such that C H ( v ) ⊆ O π ( G ) .

A new subgroup lattice characterization of finite solvable groups

We show that if G is a finite group then no chain of modular elements in its subgroup lattice L ( G ) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L ( G ) has length at least two more than the chief length of G, thereby providing a converse of a result of J. Kohler. Our results enable us to give a new characterization of finite solvable groups involving only the combinatorics of subgroup lattices. Namely, a finite group G is solvable if and only if L ( G ) contains a maximal chain X and a chain M consisting entirely of modular elements, such that X and M have the same length.