Finite Soluble Groups Research Articles

A note on conjugacy of supplements in soluble periodic linear groups

Abstract The aim of this short note is to prove that if G is a (homomorphic images of a) soluble periodic linear group and N is a locally nilpotent normal subgroup of G such that N and G / N {G/N} have no isomorphic G-chief factors, then two supplements to N in G are conjugate provided that they have the same intersection with N. This result follows from well-known theorems in the theory of Schunck classes (see [A. Ballester-Bolinches and L. M. Ezquerro, On conjugacy of supplements of normal subgroups of finite groups, Bull. Aust. Math. Soc. 89 2014, 2, 293–299]), and it appeared as the main theorem of [C. Parker and P. Rowley, A note on conjugacy of supplements in finite soluble groups, Bull. Lond. Math. Soc. 42 2010, 3, 417–419].

A generalization of an Agrawal theorem on soluble PST-groups

Let σ = { σ i ⏧ i ∈ I } be some partition of the set of all prime numbers and G a finite group. A subgroup A of G is said to be σ -permutable in G if A permutes with all Hall σi -subgroups H, that is, AH = HA for all i ∈ I . The group G is said to be a P σ T -group if σ -permutability is a transitive relation in G, that is, if K is a σ -permutable subgroup of H and H is a σ -permutable subgroup of G, then K is a σ -permutable subgroup of G. In the case when σ = σ 1 = { { 2 } , { 3 } , { 5 } , … } , a P σ T -group is called a PST-group. We study the structure of finite soluble groups G in which every subnormal subgroup is σ-permutable and G N σ ∩ H ≤ Z ∞ ( H ) for every Hall σi -subgroup H of G and all i. In particular, we obtain a new characterization of soluble P σ T -groups which is a generalization of an Agrawal theorem on soluble PST-groups.

On a question concerning the intersection of Φ-isolators of finite soluble groups

A subgroup A of a finite group G is called Φ-isolator of G if A covers the Frattini chief factors of G and avoids the supplemented ones. Let H be a Φ-isolator of a soluble finite group G. In the present paper, we prove that there exist elements x,y∈G such that the equality H∩Hx∩Hy=Φ(G) holds. This result is an answer to Question 19.38 in “The Kourovka notebook”.

On insoluble transitive subgroups in the holomorph of a finite soluble group

A question of interest both in Hopf-Galois theory and in the theory of skew braces is whether the holomorph Hol(N) of a finite soluble group N can contain an insoluble regular subgroup. We investigate the more general problem of finding an insoluble transitive subgroup G in Hol(N) with soluble point stabilisers. We call such a pair (G,N) irreducible if we cannot pass to proper non-trivial quotients G‾, N‾ of G, N so that G‾ becomes a subgroup of Hol(N‾). We classify all irreducible solutions (G,N) of this problem, showing in particular that every non-abelian composition factor of G is isomorphic to the simple group of order 168. Moreover, every maximal normal subgroup of N has index 2.

On σ-Residuals of Subgroups of Finite Soluble Groups

Let σ={σi:i∈I} be a partition of the set of all prime numbers. A subgroup H of a finite group G is said to be σ-subnormal in G if H can be joined to G by a chain of subgroups H=H0⊆H1⊆⋯⊆Hn=G where, for every j=1,⋯,n, Hj−1 is normal in Hj or Hj/CoreHj(Hj−1) is a σi-group for some i∈I. Let B be a subgroup of a soluble group G normalising the Nσ-residual of every non-σ-subnormal subgroup of G, where Nσ is the saturated formation of all σ-nilpotent groups. We show that B normalises the Nσ-residual of every subgroup of G if G does not have a section that is σ-residually critical.

Finite groups with $H_{\mathcal L}$-embedded subgroups

Let G be a finite soluble group and let \mathfrak{F} be a class of groups. A chief factor H/K of G is said to be \mathfrak{F} -central (in G ) if (H/K)\rtimes (G/C_{G}(H/K)) \in \mathfrak{F} ; we write {\mathcal L}_{c\mathfrak{F}}(G) to denote the set of all subgroups A of G such that every chief factor H/K of G between A_{G} and A^{G} is \mathfrak{F} -central in G . Let \mathcal L be a set of subgroups of G . We say that a subgroup A of G is H_{\mathcal L} -embedded in G provided A is a Hall subgroup of some subgroup E\in {\mathcal L} . In this paper, we study the structure of G under the condition that every subgroup of G is H_{\mathcal L} -embedded in G , where {\mathcal L}={\mathcal L}_{c\mathfrak{F}}(G) for some hereditary saturated formation \mathfrak{F} . Some known results are generalized.

σ-Subnormality in locally finite groups

Let σ={σj:j∈J} be a partition of the set P of all prime numbers. A subgroup X of a finite group G is σ-subnormal in G if there exists a chain of subgroupsX=X0≤X1≤…≤Xn=G such that, for each 1≤i≤n−1, Xi−1⊴Xi or Xi/(Xi−1)Xi is a σji-group for some ji∈J. Skiba [18] studied the main properties of σ-subnormal subgroups in finite groups and showed that the set of all σ-subnormal subgroups plays a very relevant role in the structure of a finite soluble group. In this paper we lay the foundation of a general theory of σ-subnormal subgroups (and σ-series) in locally finite groups. Although in finite groups, σ-subnormal subgroups form a sublattice of the lattice of all subgroups (see for instance [3]), this is no longer true for locally finite groups; in fact, the join of σ-subnormal subgroups is not always σ-subnormal, but this is the case (for example) whenever the join of subnormal subgroups is subnormal (see Theorem 3.16). We provide many criteria to determining when a subgroup is σ-subnormal starting from the much weaker concept of σ-seriality (see Section 2). These criteria are particularly useful when employed to investigate the join of σ-subnormal subgroups — we show for example that if two σ-subnormal subgroups H and K of a locally finite group G are such that HK=KH, then HK is σ-subnormal in G (see Theorem 3.15) — but they are also fit to show that on some occasions σ-seriality coincides with σ-subnormality — this is the case of linear groups (see Theorem 3.35).

FINITE GROUPS WITH ABNORMAL MINIMAL NONNILPOTENT SUBGROUPS

AbstractWe describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and $Z(G)$ is cyclic of order $|Z(G)|\in \{1, 2, 3, 4\}$ .

P-nilpotency criteria for some verbal subgroups

In this paper we address the problem of understanding when a verbal subgroup of a finite group is p-nilpotent, with p a prime, that is, when all its elements of p′-order determine a subgroup. We provide two p-nilpotency criteria, one for the terms of the lower central series of any finite group and one for the terms of the derived series of any finite soluble group, which relies on arithmetic properties related to the order of products of commutators.

Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

Let f(x) be a non-zero polynomial with integer coefficients. An automorphism φ of a group G is said to satisfy the elementary abelian identity f(x) if the linear transformation induced by φ on every characteristic elementary abelian section of G is annihilated by f(x). We prove that if a finite (soluble) group G admits a fixed-point-free automorphism φ satisfying an elementary abelian identity f(x), where f(x) is a primitive polynomial, then the Fitting height of G is bounded in terms of deg⁡(f(x)). We also prove that if f(x) is any non-zero polynomial and G is a σ′-group for a finite set of primes σ=σ(f(x)) depending only on f(x), then the Fitting height of G is bounded in terms of the number irr(f(x)) of different irreducible factors in the decomposition of f(x). These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length α(|φ|) of 〈φ〉 when deg⁡f(x) or irr(f(x)) is small in comparison with α(|φ|).

Carter and Gasch\xfctz theories beyond soluble groups

Classical results from the theory of finite soluble groups state that Carter subgroups, i.e. self-normalizing nilpotent subgroups, coincide with nilpotent projectors and with nilpotent covering subgroups, and they form a non-empty conjugacy class of subgroups, in soluble groups. This paper presents an extension of these facts to pi -separable groups, for sets of primes pi , by proving the existence of a conjugacy class of subgroups in pi -separable groups, which specialize to Carter subgroups within the universe of soluble groups. The approach runs parallel to the extension of Hall theory from soluble to pi -separable groups by Čunihin, regarding existence and properties of Hall subgroups.

Nilpotent length and system permutability

If C is a class of groups, a C-injector of a finite group G is a subgroup V of G with the property that V∩K is a C-maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschütz and B. Hartley states the existence and conjugacy of F-injectors in finite soluble groups for Fitting classes F. We shall show that for groups of nilpotent length at most 4, F-injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

AbstractWe show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

Classes of algebras and closure operations

The calculus of classes and closure operations has proved to be a useful tool in group theory and has led to a deep theory in the study of finite soluble groups. More recently, parallel theories have started to be developed in various varieties of algebras, such as Lie, Leibniz and Malcev algebras. This paper seeks to investigate the extent to which these later theories can be generalized to the variety of all non-associative algebras.

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 nilpotency of higher commutator subgroups of a finite soluble group

Let G be a finite soluble group and $$G^{(k)}$$ the kth term of the derived series of G. We prove that $$G^{(k)}$$ is nilpotent if and only if $$|ab|=|a||b|$$ for any $$\delta _k$$ -values $$a,b\in G$$ of coprime orders. In the course of the proof, we establish the following result of independent interest: let P be a Sylow p-subgroup of G. Then $$P\cap G^{(k)}$$ is generated by $$\delta _k$$ -values contained in P (Lemma 2.5). This is related to the so-called focal subgroup theorem.

A characterization of finite soluble groups

In this article, we show that a group G is soluble if and only if G has a normal subgroup N such that G/N is soluble and every subgroup of N of order 3 or 5 is weakly c-permutable in G.

Thompson-like characterization of solubility for products of finite groups

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, $n$-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups $G = AB$ with subgroups $A,\ B$ such that $\langle a, b\rangle$ is soluble for all $a \in A$ and $b \in B$. In this case, the group $G$ is said to be an $\cal S$-connected product of the subgroups $A$ and $B$ for the class $\cal S$ of all finite soluble groups. Our main theorem states that $G = AB$ is $\cal S$-connected if and only if $[A,B]$ is soluble. In the course of the proof we derive a result of own interest about independent primes regarding the soluble graph of almost simple groups.

A probabilistic version of a theorem of lászló kovács and hyo-seob sim

For a finite group group‎, ‎denote by $mathcal V(G)$ the smallest positive integer $k$ with the property that the probability of generating $G$ by $k$ randomly chosen elements is at least $1/e.$ Let $G$ be a finite soluble group‎. ‎{Assume} that for every $pin pi(G)$ there exists $G_pleq G$ such that $p$ does not divide $|G:G_p|$ and ${mathcal V}(G_p)leq d.$ Then ${mathcal V}(G)leq d+7.$‎

The Projective Character Tables of a Solvable Group 26:6×2

The Chevalley–Dickson simple group G24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.