On a Certain Class of Metabelian Groups

Let G be a finite group with an abelian normal subgroup N. When does N have a unique conjugacy class of complements in G? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals Φ ⁢ ( G ) {\Phi(G)} . In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when Φ ⁢ ( G ) = 1 {\Phi(G)=1} , these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

Division algebras occurring in simple components of group algebras of finite groups over algebraic number fields are studied. First, well-known restrictions are presented for the structure of a group that arises once no further Clifford Theory reductions are possible. For groups with these properties, a character-theoretic condition is given that forces the p-part of the division algebra part of this simple component to be generated by a predetermined p-quasi-elementary subgroup of the group, for any prime integer p. This is effectively a constructive Brauer-Witt Theorem for groups satisfying this condition. It is then shown that it is possible to constructively compute the Schur index of a simple component of the group algebra of a finite nilpotent-by-abelian group using the above reduction and an algorithm for computing Schur indices of simple algebras generated by finite metabelian groups.

Finite metabelian groups with no outer automorphisms

The description of the automorphisms of a unipotent subgroup U of a Chevalley group over a field K known earlier for charK ≠ 2, 3 (Gibbs, 1970) was completed in 1990 together with the solution of problem (1.5) from A.S. Kondrat’ev’s survey (Usp. Mat. Nauk, 1986). In the present paper, Aut U is described for the case of finitary Chevalley groups. For a Chevalley group of classical type over a finite field, it is proved that any large Abelian subgroup from U is conjugate to a normal subgroup in U. It is shown that this is not so in the general case; therefore, problem (1.6) from Kondrat’ev’s survey about large Abelian subgroups in U is reduced to listing the exceptions. The orders of these subgroups and large Abelian normal subgroups in U were listed earlier.

A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall [STACS 2009], we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H -> H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(|H|,p^d).

The irreducible modular representations, the blocks, and the defect groups of finite metabelian groups are determined. Also the dimensions of the principal indecomposable modules are computed.

In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler--Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler--Leman (WL) coloring, which we call 2-ary WL. We then show that 2-ary WL is equivalent to the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's hierarchy. Our main result is that, in the pebble game characterization, only $O(1)$ pebbles and $O(1)$ rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in $\mathsf{P}$; Babai, Codenotti, & Qiao, ICALP 2012). We actually show that $7$ pebbles and $7$ rounds suffice. In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only $7$ variables and $7$ quantifier depth.

Variants of some of the Brauer-Fowler theorems

This chapter continues the development of group theory begun in Chapter IV, the main topics being the use of generators and relations, representation theory for finite groups, and group extensions. Representation theory uses linear algebra and inner-product spaces in an essential way, and a structure-theory theorem for finite groups is obtained as a consequence. Group extensions introduce the subject of cohomology of groups.Sections 1–3 concern generators and relations. The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup. Section 1 constructs free groups in terms of words built from an alphabet and shows that free groups are characterized by a certain universal mapping property. This universal mapping property implies that any group may be defined by generators and relations. Computations with free groups are aided by the fact that two reduced words yield the same element of a free group if and only if the reduced words are identical. Section 2 obtains the Nielsen-Schreier Theorem that subgroups of free groups are free. Section 3 enlarges the construction of free groups to the notion of the free product of an arbitrary set of groups. Free product is what coproduct is for the category of groups; free groups themselves may be regarded as free products of copies of the integers.Sections 4–5 introduce representation theory for finite groups and give an example of an important application whose statement lies outside representation theory. Section 4 contains various results giving an analysis of the space C(G, ?) of all complex-valued functions on a finite group G. In this analysis those functions that are constant on conjugacy classes are shown to be linear combinations of the characters of the irreducible representations. Section 5 proves Burnside’s Theorem as an application of this theory—that any finite group of order paqb with p and q prime and with a+b > 1 has a nontrivial normal subgroup.Section 6 introduces cohomology of groups in connection with group extensions. If N is to be a normal subgroup of G and Q is to be isomorphic to G/N, the first question is to parametrize the possibilities for G up to isomorphism. A second question is to parametrize the possibilities for G if G is to be a semidirect product of N and Q.

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω0(G) ⊇ … ⊇ Ωn(G) … of closed characteristic subgroups of G with two important properties, namely, that their intersection ∩∞n=1 Ωn(G) is Z0(G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ωn−1(G)/Ωn(G), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).In the special case that G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is profinite, our theorem sharpens a theorem of Varopoulos [8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.

Probability that the commutator of two group elements is equal to a given element

For a formation 𝔉 of finite groups, consider the graph whose vertices are elements of a finite group, with two vertices connected by an edge if and only if they generate a non-𝔉-group as elements of the group. A hereditary formation 𝔉 is said to be regular if the set of all isolated vertices of the described graph coincides with the intersection of all 𝔉-maximal subgroups in every group. In this paper, a constructive description of saturated regular formations of soluble groups is given, which improves the results of Lucchini and Nemmi. In particular, it is shown that saturated regular non-empty formations of soluble groups are just hereditary formations 𝔉 of soluble groups that contain every group whose cyclic primary subgroups are all 𝐾-𝔉-subnormal. Also, we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz lattice.

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called $${\mathcal{PT}}$$ -groups. In particular, it is shown that the finite solvable $${\mathcal{PT}}$$ -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not $${\mathcal{PT}}$$ -groups but all subnormal subgroups of defect two are permutable.

In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio [Formula: see text]. This extends the case [Formula: see text] studied in previous papers [R. Brandl and W. Shi, Finite groups whose element orders are consecutive integers, J. Algebra 143 (1991) 388–400; Y. Feng, Finite groups whose abelian subgroup orders are consecutive integers, J. Math. Res. Exp. 18 (1998) 503–506; W. Shi, Finite groups whose proper subgroup orders are consecutive integers, J. Math. Res. Exp. 14 (1994) 165–166].

Expanders are highly connected sparse graphs of great interest in computer science, in areas ranging from parallel computation to complexity theory, from cryptography to coding theory, and, most recently, computational group theory (see e.g. [AKS, G+, LP, SS, V, WZ].) The explicit constructions of expander graphs [M1, M2, LPS] (see also [GG, Lu]) use deep mathematical tools to construct families of Cayley graphs of finite groups. The Independence Problem [LW], is whether being an expander family is a property of the groups alone, independent of the choice of generators. A counterexample to the general the problem was recently obtained in [ALW], by using a new combinatorial construction of expanders [RVW]. The problem remains for all classical series of finite simple groups (An, PSL(2,p), etc.)Let G be a finite group generated by at most d elements. The product replacement graph Γk(G) is defined to be a graph, with vertices corresponding to generating k-tuples in G, and edges corresponding to Nielsen transformations. Most recently, graphs Γk(G) became a subject of an intense investigation, prompted by the study of a commonly used 'practical' product replacement algorithm for generating random elements in finite groups, designed Leedham-Green and Soicher [LG]. This algorithm, based on the random walk on graphs Γk(G), showed a remarkable performance, as reported in [C+]. It was suggested in [LP], and proved in several special cases, that the product replacement graphs Γk(G) are expanders, for a fixed k, when |G| → ∞The main result of this paper is Theorem 2, establishing the connection between the expansion coefficient of the product replacement graph Γk(G) and the minimal expansion coefficient of a Cayley graph of group G with k generators. One can think of our result as of an additional evidence in favor of the speculation in [LP]. On the other hand, it gives an algorithmic motivation for study of the Independence Problem, in the aftermath of [ALW].In a special case (see Theorem 1 below), we show that if one assumes that all Cayley graphs with at most four generators in PSL (2, p) have a universal lower bound on expansion, then the product replacement graphs Γk(PSL(2,p)) form an expander family, when k ≥ 8 is fixed, and p → ∞.The proof of the Theorems consists of two parts: probability on groups and graph theoretic. The latter is based on a new combinatorial result of independent interest (Lemmas 4, 5), which generalizes a classical lemma that the direct product of expanders is an expander (see e.g. [Ch]). We prove that if a graph Γ on a direct product of sets satisfies two conditions:1) all 'row' subgraphs are expanders,2) at least (1-e) proportion of 'column' graphs are expanders,then graph &Gamma is also an expander. We use a novel combinatorial technique based on graph decomposition, as opposed to path arguments used in previously in [CG, DS2, DS3, P4]. It is easy to see that such a technique can never prove that a certain family of graphs is an expander family (cf. [P3]).