Chief Series Research Articles

On the partial Π-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups

Let H be a subgroup of a finite group G. We say that H satisfies the partial Π -property in G if there exists a chief series Γ G : 1 = G 0 < G 1 < · · · < G n = G of G such that for every G-chief factor G i / G i − 1 ( 1 ≤ i ≤ n ) of Γ G , | G / G i − 1 : N G / G i − 1 ( H G i − 1 / G i − 1 ∩ G i / G i − 1 ) | is a π ( H G i − 1 / G i − 1 ∩ G i / G i − 1 ) -number. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial Π -property on the structure of a finite group.

On semi-cover-avoiding 2-maximal subgroups of finite groups

A subgroup $H$ of a finite group $G$ is said to be ``semi-cover-avoiding in $G$'', if there exists a chief series of $G$ such that $H$ covers or avoids every chief factor of the chief series. In this article, we will consider some 2-maximal subgroups with the property of semi-cover-avoiding of a group $G$ and explore the structure of $G$.

The minimum generating set problem

Let G be a finite group. In order to determine the smallest cardinality d(G) of a generating set of G and a generating set with this cardinality, one should repeat ‘many times’ the test whether a subset of G of ‘small’ cardinality generates G. We prove that if a chief series of G is known, then the numbers of these ‘generating tests’ can be drastically reduced. At most |G|13/5 subsets must be tested. This implies that the minimum generating set problem for a finite group G can be solved in polynomial time.

A Jordan–Hölder theorem for skew left braces and their applications to multipermutation solutions of the Yang–Baxter equation

Skew left braces arise naturally from the study of non-degenerate set-theoretic solutions of the Yang–Baxter equation. To understand the algebraic structure of skew left braces, a study of the decomposition into minimal substructures is relevant. We introduce chief series and prove a strengthened form of the Jordan–Hölder theorem for finite skew left braces. A characterization of right nilpotency and an application to multipermutation solutions are also given.

Orbit sizes and odd order composition factors of finite linear groups

In this paper, we study the existence of large orbits of odd-order subgroups of a finite solvable linear group G, and present some applications on bounding the odd-order parts of the chief series of an arbitrary linear group G. We also find applications to Gluck's conjecture as well as a related question of Navarro.

A Jordan–Hölder type theorem for supercharacter theories

Abstract The Jordan–Hölder theorem is a general term given to a collection of theorems about maximal chains in suitably nice lattices. For example, the well-known Jordan–Hölder type theorem for chief series of finite groups has been rather useful in studying the structure of finite groups. In this paper, we present a Jordan–Hölder type theorem for supercharacter theories of finite groups, which generalizes the one for chief series of finite groups.

Dense normal subgroups and chief factors in locally compact groups

In The essentially chief series of a compactly generated locally compact group, an analogue of chief series for finite groups is discovered for compactly generated locally compact groups. In the present article, we show that chief factors necessarily exist in all locally compact groups with sufficiently rich topological structure. We also show that chief factors have one of seven types, and for all but one of these types, there is a decomposition into discrete groups, compact groups, and topologically simple groups. Our results for chief factors require exploring the theory developed in Chief factors in Polish groups in the setting of locally compact groups. In this context, we obtain tighter restrictions on the factorization of normal compressions and the structure of quasi-products. Consequently, both (non-)amenability and elementary decomposition rank are preserved by normal compressions.

On the Chief Factors of Parabolic Maximal Subgroups of Special Finite Simple Groups of Exceptional Lie Type

Considering the finite simple groups F4(2 n ) and G2(p n ), where p ≤ 3, we give a description of the chief factors of a parabolic maximal subgroup involved in its unipotent radical. For every parabolic maximal subgroup of the groups F4(2 n ), G2(2 n ), and G2(3 n ), we give the fragment of its chief series that involves in the unipotent radical of this parabolic subgroup. The generators of the corresponding chief factors are presented in three tables.

The essentially chief series of a compactly generated locally compact group

We first obtain finiteness properties for the collection of closed normal subgroups of a compactly generated locally compact group. Via these properties, every compactly generated locally compact group admits an essentially chief series – i.e. a finite normal series in which each factor is either compact, discrete, or a topological chief factor. A Jordan–Holder theorem additionally holds for the ‘large’ factors in an essentially chief series.

On partial CAP∗-subgroups of finite groups

A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be a partial [Formula: see text]-subgroup of [Formula: see text] if there exists a chief series [Formula: see text] of [Formula: see text] such that [Formula: see text] either covers or avoids each non-Frattini chief factor of [Formula: see text]. In this paper, we study the influence of the partial [Formula: see text]-subgroups on the structure of finite groups. Some new characterizations of the hypercyclically embedded subgroups, [Formula: see text]-nilpotency and supersolubility of finite groups are obtained.

SCAP-subalgebras of Lie algebras

A subalgebra H of a finite dimensional Lie algebra L is said to be a SCAP-subalgebra if there is a chief series 0 = L 0 ⊂ L 1 ⊂... ⊂ L t = L of L such that for every i = 1, 2,..., t, we have H + L i = H + L i-1 or H ∩ L i = H ∩ L i-1. This is analogous to the concept of SCAP-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its SCAP-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

On finite groups with some primary subgroups satisfying partial S-Π-property

ABSTRACTA p-subgroup H of a finite group G is said to satisfy partial S-Π-property in G if G has a chief series such that for every G-chief factor (1≤i≤n) of ΓG, either is a Sylow p-subgroup of or is a p-number. In this paper, we mainly investigate the structure of finite groups with some primary subgroups satisfying partial S-Π-property.

On maximal subalgebras and a generalized Jordan–Hölder theorem for Lie algebras

ABSTRACTThe purpose of this paper is to continue the study of chief factors of a Lie algebra and to prove a further strengthening of the Jordan–Hölder theorem for chief series.

On the cohomology of p-solvable groups

Suppose V is an irreducible \({\mathbb F}_pG\)-module where p is a prime and G is a finite p-solvable group. Then by a result of Gaschutz \(\dim _FH^1(G,V)=s_G(V)\) where \(F=\text {End}_G(V)\) and \(s_G(V)\) is the multiplicity of V as a split chief factor in any chief series of G. One also knows that \(\dim _FH^1(G,V)\le \dim _FH^2(G,V)\), and the object of the present paper is to explain this inequality in terms of Gaschutz’s result.

On the chief factors of maximal parabolic subgroups of twisted classical groups

For the finite simple groups of twisted Lie types 2Al and 2Dl, we specify the description for the chief factors of a maximal parabolic subgroup which are involved in its unipotent radical. We prove a theorem in which, for every maximal parabolic subgroup of the groups 2Al(q2) and 2Dl(q2), we give the fragments of the chief series involving in the unipotent radical of this parabolic subgroup. The generators of the corresponding chief factors are presented in tables.

On chief factors of parabolic maximal subgroups of the group 2 E 6(q 2)

For a finite simple group of twisted Lie type 2E6, the description of chief factors of a parabolic maximal subgroup contained in its unipotent radical is refined. The results are presented as a theorem, in which, for every parabolic maximal subgroup of 2E6(q2), the fragments of chief series contained in the unipotent radical of this parabolic subgroup are given. Generating elements and orders of the corresponding chief factors are presented in a table.

On partial CAP-subgroups of finite groups

Given a chief factor H/K of a finite group G, we say that a subgroup A of G avoids H/K if H∩A=K∩A; if HA=KA, then we say that A covers H/K. If A either covers or avoids the chief factors of some given chief series of G, we say that A is a partial CAP-subgroup of G. Assume that G has a Sylow p-subgroup of order exceeding pk. If every subgroup of order pk, where k≥1, and every subgroup of order 4 (when pk=2 and the Sylow 2-subgroups are non-abelian) are partial CAP-subgroups of G, then G is p-soluble of p-length at most 1.

Groups of Induced Automorphisms and Their Application to Studying the Existence Problem for Hall Subgroups

In the present paper, the term a ‘group’ will always refer to a finite group. By writing A G and A G we mean that A is, respectively, a subgroup and a normal subgroup of G. An unrefinable normal series is called a chief series. A composition series is called an (rc)-series if it is a ramification of a chief series [1]. Let A, B, and H be subgroups of G such that B A. The normalizer of the section A/B in H is defined thus: NH(A/B) := NH(A) ∩ NH(B). If x ∈ NH(A/B), then x induces an automorphism on A/B acting by the rule Ba → Bx−1ax. Thus there exists a homomorphism NH(A/B) → Aut (A/B). The image of the normalizer NH(A/B) under this automorphism is denoted by AutH(A/B) and is called the group of H-induced automorphisms of A/B; the kernel of the homomorphism is denoted by CH(A/B). If B = 1, then we use the designation AutH(A). Note that in the literature AutG(A) is sometimes called the automizer of a subgroup A in a group G. Groups of induced automorphisms first appeared in [1] where is was not mentioned that this notion had been borrowed from unpublished lectures of H. Wielandt. Obviously, CH(A/B) = CG(A/B) ∩H, and hence

A practical model for computation with matrix groups

We describe an algorithm to compute a composition tree for a matrix group defined over a finite field, and show how to use the associated structure to carry out computations with such groups; these include finding composition and chief series, the soluble radical, and Sylow subgroups.

On the Lengths of Certain Chains of Subalgebras in Lie Algebras

In this paper we study the lengths of certain chains of subalgebras of a Lie algebra L: namely, a chief series, a maximal chain of minimal length, a chain of maximal length in which each subalgebra is modular in L, and a chain of maximal length in which each subalgebra is a quasi-ideal of L. In particular we show that, over a field F of characteristic zero, a Lie algebra L with radical R has a maximal chain of subalgebras and a chain of subalgebras all of which are modular in L of the same length if and only if L = R, or and L/R is a direct sum of isomorphic three-dimensional simple Lie algebras.