Finite Solvable Groups Research Articles

A new criterion for p-nilpotency and solvability of finite groups by coprime actions

Let G and A be finite groups of relative coprime orders and A acts on G via automorphisms. In this paper, we investigate the p-nilpotency and solvability of finite groups in which every maximal A-invariant subgroup of a Sylow p-subgroup of G for some prime p is S-semipermutable.

On Gluck’s conjecture for wreath product type groups

Abstract A well-known conjecture of Gluck claims that | G : F ( G ) | ≤ b ( G ) 2 \lvert G:\mathbf{F}(G)\rvert\leq b(G)^{2} for all finite solvable groups 𝐺, where F ⁢ ( G ) \mathbf{F}(G) is the Fitting subgroup and b ⁢ ( G ) b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G ≀ H 1 ≀ H 2 ≀ ⋯ ≀ H r G\wr H_{1}\wr H_{2}\wr\cdots\wr H_{r} , where 𝐺 is a finite solvable group acting primitively on F ⁢ ( G ) / Φ ⁢ ( G ) \mathbf{F}(G)/\Phi(G) , and each H i H_{i} is a solvable primitive permutation group of finite degree.

On the number of conjugacy classes of a finite solvable group

On the number of conjugacy classes of a finite solvable group

Classifying primitive solvable permutation groups of rank 5 and 6

Abstract Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let G 0 G_{0} be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of G 0 G_{0} in Ω, including the trivial orbit { α } \{\alpha\} . In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.

The pro-supersolvable topology on a free group: Deciding denseness

Let F be a free group of arbitrary rank and let H be a finitely generated subgroup of F. Given a pseudovariety V of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow F with its pro-V topology. Our main result states that it is decidable whether H is pro-Su dense, where Su⊂S denote respectively the pseudovarieties of all finite supersolvable groups and all finite solvable groups. Our motivation stems from the following open problem: is it decidable whether H is pro-S dense?

Galois Conjugacy and Fitting Height 2: Classification of Finite Solvable Groups

This paper investigates the finite groups 𝐺 for which any two characters in the set of irreducible complex characters 𝑰𝒓𝒓(𝐺) are Galois conjugate. Specifically, we classify such groups and establish a key result: they are solvable with Fitting height 2. The analysis involves intricate considerations of irreducible complex characters and their Galois conjugacy, shedding light on the structural properties of finite solvable groups.

On the Orbital Regular Graph of Finite Solvable Groups

Let G be a finite solvable group and Δ be the subset of Υ×Υ, where Υ is the set of all pairs of size two commuting elements in G. If G operates on a transitive G – space by the action (υ1,υ2)g=(υg1,υg2); υ1,υ2∈Υ and g∈G, then orbits of G are called orbitals. The subset Δo={(υ,υ);υ∈Υ,(υ,υ)∈Υ×Υ} represents G′s diagonal orbital.The orbital regular graph is a graph on which G acts regularly on the vertices and the edge set. In this paper, we obtain the orbital regular graphs for some finite solvable groups using a regular action. Furthermore, the number of edges for each of a group’s orbitals is obtained.

SOLVABLE GROUPS WHOSE NONNORMAL SUBGROUPS HAVE FEW ORDERS

Abstract Suppose that G is a finite solvable group. Let $t=n_c(G)$ denote the number of orders of nonnormal subgroups of G. We bound the derived length $dl(G)$ in terms of $n_c(G)$ . If G is a finite p-group, we show that $|G'|\leq p^{2t+1}$ and $dl(G)\leq \lceil \log _2(2t+3)\rceil $ . If G is a finite solvable nonnilpotent group, we prove that the sum of the powers of the prime divisors of $|G'|$ is less than t and that $dl(G)\leq \lfloor 2(t+1)/3\rfloor +1$ .

The Gruenberg-Kegel graph of finite solvable rational groups

A finite group G is said to be rational if every character of G is rational-valued. The Gruenberg-Kegel graph of a finite group G is the undirected graph whose vertices are the primes dividing the order of G and an edge connects a pair of different vertices p and q whenever G has an element of order pq. In this paper, we complete the classification of the Gruenberg-Kegel graphs of finite solvable rational groups initiated in [1].

Finite Solvable Groups in Which the $$\sigma$$-Quasinormality of Subgroups is a Transitive Relation

Finite Solvable Groups in Which the $$\sigma$$-Quasinormality of Subgroups is a Transitive Relation

Uniform cyclic group factorizations of finite groups

In this paper, we introduce a kind of decomposition of a finite group called a uniform group factorization, as a generalization of exact factorizations of a finite group. A group G is said to admit a uniform group factorization if there exist subgroups H 1 , H 2 , … , H k such that G = H 1 H 2 ⋯ H k and the number of ways to represent any element g ∈ G as g = h 1 h 2 ⋯ h k ( h i ∈ H i ) does not depend on the choice of g. Moreover, a uniform group factorization consisting of cyclic subgroups is called a uniform cyclic group factorization. First, we show that any finite solvable group admits a uniform cyclic group factorization. Second, we show that whether all finite groups admit uniform cyclic group factorizations or not is equivalent to whether all finite simple groups admit uniform group factorizations or not. Lastly, we give some concrete examples of such factorizations.

Erratum and Addendum: Finite solvable groups with a nilpotent normal complement subgroup

Erratum and Addendum: Finite solvable groups with a nilpotent normal complement subgroup

Orthogonal irreducible representations of finite solvable groups in odd dimension

AbstractWe prove that if is a finite irreducible solvable subgroup of an orthogonal group with odd, then preserves an orthogonal decomposition of into 1‐spaces. In particular is monomial. This generalizes a theorem of Rod Gow.

The finite and solvable genus of finitely generated free and surface groups

Let {mathcal {C}} be the pseudovariety {mathcal {F}} of all finite groups or the pseudovariety {mathcal {S}} of all finite solvable groups and let Gamma be either a finitely generated free group or a surface group. The {mathcal {C}}-genus of Gamma , denoted by {mathcal {G}}_{{mathcal {C}}}(Gamma ), consists of the isomorphism classes of finitely generated residually-mathcal C groups G having the same quotients in {mathcal {C}} as Gamma . We show that the groups from {mathcal {G}}_{{mathcal {C}}}(Gamma ) are residually-p for all primes p. This answers a question of Gilbert Baumslag and shows that the groups in the genus are residually finite rationally solvable groups. This leads to a positive solution of particular case of a question of Alexander Grothendieck: if F is a free group, G is a finitely generated residually-{mathcal {C}} group and u:Frightarrow G is a homomorphism such that the induced map of pro-{mathcal {C}} completions u_{widehat{{mathcal {C}}}} : F_{widehat{{mathcal {C}}}}rightarrow G_{widehat{{mathcal {C}}}} is an isomorphism, then u is an isomorphism.

Finite solvable groups with a nilpotent normal complement subgroup

Let [Formula: see text] be a finite solvable group and [Formula: see text] a non-normal core-free solvable subgroup of [Formula: see text]. We show that if the normalizer of any nontrivial normal subgroup of [Formula: see text] is equal to [Formula: see text], then [Formula: see text] has a nilpotent normal complement [Formula: see text] such that [Formula: see text] and [Formula: see text] is a Frobenius group.

New bounds for numbers of primes in element orders of finite groups

AbstractLet denote the maximal number of different primes that may occur in the order of a finite solvable group G, all elements of which have orders divisible by at most n distinct primes. We show that for all . As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups.

Irredundant families of maximal subgroups of finite solvable groups

Irredundant families of maximal subgroups of finite solvable groups

Centralizers in finite solvable groups

Centralizers in finite solvable groups

FINITE SOLVABLE TIDY GROUPS ARE DETERMINED BY HALL SUBGROUPS WITH TWO PRIMES

AbstractIn this paper, we investigate finite solvable tidy groups. We prove that a solvable group with order divisible by at least two primes is tidy if all of its Hall subgroups that are divisible by only two primes are tidy.

Solvable groups with four conjugacy classes outside a normal subgroup

Abstract In this paper, we investigate the structure of finite solvable groups G admitting a normal subgroup N such that G / N {G/N} is non-abelian and N contains all but four conjugacy classes of G.