Soluble Subgroup Research Articles

Nonsoluble and non-p-soluble length of finite groups

Every finite group G has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length λ(G) as the number of nonsoluble factors in a shortest series of this kind. Upper bounds for λ(G) appear in the study of various problems on finite, residually finite, and profinite groups. We prove that λ(G) is bounded in terms of the maximum 2-length of soluble subgroups of G, and that λ(G) is bounded by the maximum Fitting height of soluble subgroups. For an odd prime p, the non-p-soluble length λ p (G) is introduced, and it is proved that λ p (G) does not exceed the maximum p-length of p-soluble subgroups. We conjecture that for a given prime p and a given proper group variety V the non-p-soluble length λ p (G) of finite groups G whose Sylow p-subgroups belong to V is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent. As an application of the results obtained, an error is corrected in the proof of the main result of the second author’s paper Multilinear commutators in residually finite groups, Israel Journal of Mathematics 189 (2012), 207–224.

Locally finite groups containing a $$2$$ 2 -element with Chernikov centralizer

Suppose that a locally finite group $$G$$ has a 2-element $$g$$ with Chernikov centralizer. It is proved that if the involution in $$\langle g\rangle $$ has nilpotent centralizer, then $$G$$ has a soluble subgroup of finite index.

Growth in finite simple groups of Lie type

We prove that ifLLis a finite simple group of Lie type andAAa set of generators ofLL, then eitherAAgrows, i.e.,|A3|>|A|1+ε|A^3| > |A|^{1+\varepsilon }whereε\varepsilondepends only on the Lie rank ofLL, orA3=LA^3=L. This implies that for simple groups of Lie type of bounded rank a well-known conjecture of Babai holds, i.e., the diameter of any Cayley graph is polylogarithmic. We also obtain new families of expanders.A generalization of our proof yields the following. LetAAbe a finite subset ofSL(n,F)SL(n,\mathbb {F}),F\mathbb {F}an arbitrary field, satisfying|A3|≤K|A|\big |A^3\big |\le \mathcal {K}|A|. ThenAAcan be covered byKm\mathcal {K}^m, i.e., polynomially many, cosets of a virtually soluble subgroup ofSL(n,F)SL(n,\mathbb {F})which is normalized byAA, wheremmdepends onnn.

Power sets and soluble subgroups

We prove that for certain positive integers k, such as 12, a normal subgroup of a finite group which consists of k powers is necessarily soluble. This gives rise to new solubility criteria, and solves an open problem from [2].

On the definability of radicals in supersimple groups

AbstractIf G is a group with a supersimple theory having a finite SU-rank, then the subgroup of G generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If H is any group with a supersimple theory, then the subgroup of H generated by all of its normal soluble subgroups is definable and soluble.

A remark on the definability of the Fitting subgroup and the soluble radical

AbstractLet G be an arbitrary group. We show that if the Fitting subgroup of G is nilpotent then it is definable. We prove also that the class of groups whose Fitting subgroup is nilpotent of class at most n is elementary. We give an example of a group (arbitrary saturated) whose Fitting subgroup is definable but not nilpotent. Similar results for the soluble radical are given; that is for the subgroup generated by all normal soluble subgroups of G.

Chromatic number and clique number of subgraphs of regular graph of matrix algebras

Let R be a ring and X⊆R be a non-empty set. The regular graph of X, Γ(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Γ(GLn(F)) finite? In this paper, we show that if G is a soluble subgroup of GLn(F), then χ(Γ(G))<∞. Also, we show that for every field F, χ(Γ(Mn(F)))=χ(Γ(Mn(F(x)))), where x is an indeterminate. Finally, for every algebraically closed field F, we determine the maximum value of the clique number of Γ(<A>), where <A> denotes the subgroup generated by A∈GLn(F).

Saturated formations and products of connected subgroups

For a non-empty class of groups C , two subgroups A and B of a group G are said to be C -connected if 〈 a , b 〉 ∈ C for all a ∈ A and b ∈ B . Given two sets π and ρ of primes, S π S ρ denotes the class of all finite soluble groups that are extensions of a normal π-subgroup by a ρ-group. It is shown that in a finite group G = A B , with A and B soluble subgroups, then A and B are S π S ρ -connected if and only if O ρ ( B ) centralizes A O π ( G ) / O π ( G ) , O ρ ( A ) centralizes B O π ( G ) / O π ( G ) and G ∈ S π ∪ ρ . Moreover, if in this situation A and B are in S π S ρ , then G is in S π S ρ . This result is then extended to a large family of saturated formations F , the so-called nilpotent-like Fitting formations of soluble groups, and to finite groups that are products of arbitrarily many pairwise permutable F -connected F -subgroups.

Finite Soluble Group Generated by the Conjugacy Class of an Involution

Let G be a finite group and P a subgroup of order 2. We study in this article the structures of the soluble subgroup of G that is generated by three conjugates of P. We use the results we proved about the soluble subgroups that are generated by three conjugates of P to find a soluble analogue of the Baer–Suzuki Theorem in the case prime 2.

Linear groups with many two-generator soluble subgroups

Let G be a linear group of degree n over a field and let S be any generating set. It is proved that (a) if every pair of products of at most ( n ! ) 3 n 2 elements of S ∪ S− 1 generates a soluble group then G is soluble and (b) if G is soluble and every pair of products of at most n 3 + log 2 n elements of S ∪ S−1 generates a nilpotent group then G is locally nilpotent.

Soluble products of connected subgroups

The main result in the paper states the following: For a finite group G=AB , which is the product of the soluble subgroups A and B , if \langle a,b \rangle is a metanilpotent group for all a\in A and b\in B , then the factor groups \langle a,b \rangle F(G)/F(G) are nilpotent, F(G) denoting the Fitting subgroup of G . A particular generalization of this result and some consequences are also obtained. For instance, such a group G is proved to be soluble of nilpotent length at most l+1 , assuming that the factors A and B have nilpotent length at most l . Also for any finite soluble group G and k\geq 1 , an element g\in G is contained in the preimage of the hypercenter of G/F_{k-1}(G) , where F_{k-1}(G) denotes the ( k-1 )th term of the Fitting series of G , if and only if the subgroups \langle g,h\rangle have nilpotent length at most k for all h\in G .

On the Fitting height of a soluble group that is generated by a conjugacy class of 3-elements (II)

Let G be a fnite group and P a subgroup of order 3. In this paper we proved some results about the soluble subgroups generated by three conjugates of P and we use these results to produce some properties of the group G.

ON THE CENTRALIZER OF AN ELEMENT OF ORDER FOUR IN A LOCALLY FINITE GROUP

AbstractWe prove that if G is a locally finite group admitting an automorphism φ of order four such that CG(φ) is Chernikov, then G has a soluble subgroup of finite index.

Stable pseudofinite groups

The main theorem is that if G is a pseudofinite group with stable theory, then G has a definable normal soluble subgroup of finite index.

Finite groups with few non-cyclic subgroups

For a group H we denote by δ(H) the number of conjugacy classes of non-cyclic subgroups. Groups all of whose soluble subgroups H satisfy δ(H) ≤ 2 are classified. It is also shown that the derived length and Fitting length of a soluble group G are bounded by functions of δ(G).

On soluble skew linear groups over finite-dimensional division algebras

Let D be a division ring of finite degree d and let n be a positive integer. If G is any soluble subgroup of GL ( n , D ) , we prove that G has derived length at most 9 + log 2 d + ( 11 / 3 ) log 2 n and that G has a unipotent-by-abelian (abelian if G is completely reducible) normal subgroup of finite index dividing b ( n ) . d 2 n , where b ( n ) is an integer-valued function of n only. Actually, we derive bounds rather better than those quoted above, but rather more involved to state.

The probability of generating a soluble subgroup of a finite group

It is proved that for finite groups G, the probability that two randomly chosen elements of G generate a soluble subgroup tends to zero as the index of the largest soluble normal subgroup of G tends to infinity.

LINEAR GROUPS OF SMALL DEGREE OVER FINITE FIELDS

For n = 2,3 and finite field 𝔼 of characteristic greater than n, we provide a complete and irredundant list of soluble irreducible subgroups of GL (n,𝔼). The insoluble irreducible subgroups of GL (2,𝔼) are similarly determined. Each group is given explicitly by a generating set of matrices. The lists are available electronically.

Crossed product conditions for division algebras of prime power degree

Let D be an F-central division algebra of degree p r , p a prime. A set of criteria is given for D to be a crossed product in terms of irreducible soluble or abelian-by-finite subgroups of the multiplicative group D * of D. Using the Amitsur's classification of finite subgroups of D * and the Tits alternative, it is shown that D is a crossed product if and only if D * contains an irreducible soluble subgroup. Further criteria are also presented in terms of irreducible abelian-by-finite subgroups and irreducible subgroups satisfying a group identity. Using the above results, it is shown that if D * contains an irreducible finite subgroup, then D is a crossed product.

An Analog for the Frattini Factorization of Finite Groups

Using the classification of finite simple groups, we prove that if H is an insoluble normal subgroup of a finite group G, then H contains a maximal soluble subgroup S such that G=HNG(S). Thereby Problem 14.62 in the “Kourovka Notebook” is given a positive solution. As a consequence, it is proved that in every finite group, there exists a subgroup that is simultaneously a \({\mathfrak{S}}\)-projector and a \({\mathfrak{S}}\)-injector in the class, \({\mathfrak{S}}\), of all soluble groups.