Solubility Of Finite Groups Research Articles

On $\sigma$-$c$-subnormal subgroups of finite groups

Let $ \sigma=\{\sigma_i:i\in I\} $ be a partition of the set $ \mathbb{P} $ of all primes. A finite group $ G $ is called $ \sigma $-primary if the prime divisors, if any, of $|G|$ all belong to the same member of $ \sigma $. A finite group $ G $ is called $ \sigma $-soluble if every chief factor of $ G $ is $ \sigma$-primary. A subgroup $H$ of a group $G$ is called $\sigma$-subnormal in $G$ if there is a chain of subgroups $H=H_0\leq H_1\leq\cdots\leq H_n=G$ such that either $ H_{i-1} $ is normal in $ H_i $ or $ H_{i}/(H_{i-1})_{H_{i}} $ is $ \sigma $-primary for all $ i=1,\dots,n $; A subgroup $H$ of a group $G$ is called $\sigma$-$c$-subnormal in $G$ if there is a subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq H_{\sigma G}$, where the subgroup $H_{\sigma G}$ is generated by all $\sigma$-subnormal subgroups of $G$ contained in $H$. In this paper, we investigate the influence of $\sigma$-$c$-subnormality of some kinds of maximal
 subgroups on $\sigma$-solubility of finite groups, which generalize some known results.

Second Maximal Invariant Subgroups and Solubility of Finite Groups

Second Maximal Invariant Subgroups and Solubility of Finite Groups

Products of Finite Connected Subgroups

For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if ⟨a,b⟩∈L for all a∈A and b∈B. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.

On sse-embedded subgroups of finite groups

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.

A characterization of solubility of finite groups

A subgroup H of a group G is said to be G-semipermutable in G if H has a supplement T in G such that $$G=HT$$ and for every subgroup $$T_1$$ of T, $$HT_1^g=T_1^gH$$ for some element g in G. In this paper, we investigate the structure of G under the assumption that a soluble maximal subgroup of G is G-semipermutable in G.

On F<sub>τ</sub>-s-supplemented Subgroups of Finite Groups

Let F be a non-empty formation of groups, τ a subgroup functor and H a p-subgroup of a finite group G. Let Ḡ=G/HG and H = H/HG. We say that H is Fτs-supplemented in G if for some subgroup T and some τ-subgroup S of Ḡ contained in H, HT is subnormal in Ḡ and H∩ T ≤ SZF(Ḡ). In this paper, we investigate the influence of Fτ-s-supplemented subgroups on the structure of finite groups. Some new characterizations about solubility of finite groups are obtained. AMS subject classifications: 20D10, 20D15, 20D20

On p-supersolubility and solubility of finite groups

Let 𝔉 be a class of groups. A subgroup H of G is said to be weakly 𝔉s-quasinormal in G if G has an S-quasinormal subgroup T such that HT is S-quasinormal in G and (H ∩ T) HG/HG ≤ Z𝔉(G/HG), where Z𝔉(G/HG) is the 𝔉-hypercenter of G/HG. In this paper, we investigate further the influence of weakly 𝔉s-quasinormality of some subgroups on the structure of finite groups. Some new characterizations about p-supersolubility and solubility of finite groups are obtained.

ON A THEOREM OF HUPPERT

A well-known theorem of Huppert states that a finite group is soluble if its every proper subgroup is supersoluble. In this paper, we proved the following result: let G be a finite group. (1) If G has exactly n non-supersoluble proper subgroups, where 0 ≤ n ≤ 7 and n ≠ 5, then G is soluble. (2) G is a non-soluble group with exactly five non-supersoluble proper subgroups if and only if all non-supersoluble proper subgroups are conjugate maximal subgroups and G/Φ(G) ≅ A5, where Φ(G) is the Frattini subgroup of G. Furthermore, we also considered the influence of the number of non-abelian proper subgroups on the solubility of finite groups.

A note on p-nilpotence and solvability of finite groups

In this note, we first give some examples to show that some hypotheses of some well-known results for a group G to be p-nilpotent, solvable and supersolvable are essential and cannot be removed. Second, we give some generalizations of two theorems in [A. Ballester-Bolinches, X. Guo, Some results on p-nilpotence and solubility of finite groups, J. Algebra 228 (2000) 491–496].

Some Results on p-Nilpotence and Solubility of Finite Groups

Some Results on p-Nilpotence and Solubility of Finite Groups

Solubility of finite groups admitting a certain fixed-point-free group of automorphisms

Solubility of finite groups admitting a certain fixed-point-free group of automorphisms

Solubility of finite groups admitting a fixed-point-free automorphism of orderrst III

This paper, which is one of a series of four, contributes to the proof of the following Theorem.A finite group admitting a coprime fixed-point-free automorphism α of order rst (r, s andt distinct primes)is soluble. Here we prove that in a minimal counterexample to the above theorem the set ofα-invariant Sylowp-subgroupsP, such thatC p(α i)≠1 for allα i≠1, generate a soluble subgroup.

Solubility of finite groups admitting a fixed-point-free automorphism of order rst, II

This paper, which is one of a series of four, contributes to the proof of the following THEOREM. A finite group admitting a coprime fixed-point-free automorphism a of order rst (r, s and t distinct primes) is soluble. Here we prove that in a minimal counterexample to the above theorem the set of a-invariant Sylow p-subgroups P, such that Cp(~')~l for all c~l, generate a soluble subgroup.

Solubility of Finite Groups Admitting a Fixed-Point-Free Abelian Automorphism Group of Square-Free Exponent rs

Solubility of Finite Groups Admitting a Fixed-Point-Free Abelian Automorphism Group of Square-Free Exponent <i>rs</i>