On 9- and 10-decomposable finite groups

A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. In some research papers, the question of finding all positive integer n such that there is an n-decomposable finite group was posed. In this paper, we investigate the structure of 9- and 10-decomposable non-perfect finite groups. We prove that a non-perfect group G is 9-decomposable if and only if G is isomorphic to Aut(PSL(2,32)), Aut(PSL(3,3)), the semi-direct product Z ₃ ∝(Z ₅×Z ₅) or a non-abelian group of order pq, where p and q are primes and p?1=8q, and also, a non-perfect finite group G is 10-decomposable if and only if G is isomorphic to Aut(PSL(2,17)), PSL(2,25):2₃, a split extension of PSL(2,25) by Z ₂ in ATLAS notation (Conway et al., Atlas of Finite Groups, [1985]), Aut(U ₃(3)) or D ₃₈, where D ₃₈ denotes the dihedral group of order 38.     

Characterization of classes of finite groups with the use of generalized subnormal Sylow subgroups

A characterization of the classes of all π-nilpotent, all π-closed, and all π-decomposable finite groups is obtained by using generalized subnormal Sylow subgroups.     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.     

Spectral analysis of certain compact factors for Gaussian dynamical systems

For factors of a Gaussian automorphismT determined by compact subgroups of the group of unitary operators acting onL ² of the spectral measure ofT, we prove that the maximal spectral multiplicity is either 1 or infinity. As an application, we show that the maximal multiplicity of those factors an allL ^p, 1<p<+∞, is the same.     

On the duality theorem of bounded S-decomposable operators

A class of linear operators, more general than that of the decomposable operators, here referred to as S-decomposable operators, was introduced in [10]. It was shown in [11] that the dual of an S-decomposable operator was again S-decomposable. Our main result, in this paper, is the converse implication, namely, if the dual $\text{T1}$ of a bounded linear operator T is S-decomposable then T is S-decomposable.     

Product of a biprimary and A 2-decomposable group

Suppose a finite group G is the product of a subgroups A and B of coprime orders, and suppose the order of A is p ^a q^b, where p and q are primes, and B is a 2-decomposable group of even order. Assume that a Sylow p-subgroup P is cyclic. If the order of P is not equal to 3 or 7, then G is solvable. If G is nonsolvable and G contains no nonidentity solvable invariant subgroups, then G is isomorphic to PSL(2, 7) or PGL(2, 7).Translated from Matematicheskie Zametki, Vol. 23, No. 5, pp. 641–649, May, 1978.     

On decomposability of finite groups

Let G be a finite group. A normal subgroup N of G is a union of several G-conjugacy classes, and it is called n-decomposable in G if it is a union of n distinct G-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.     

Additive properties of subgroups of finite index in fields

In this paper, we confirm a conjecture of Bergelson and Shapiro concerning subgroups of finite index in multiplicative groups of fields which have maximal additive dimension. We also show that the natural extension of subgroups Gp of prime index p inside Q^? and additive dimension p+1 to the case where p is replaced by a composite integer n leads to subgroups of bounded additive dimension on a set of positive integers n of asymptotic density 1.     

On <Emphasis Type="Italic">p</Emphasis>-nilpotence and solubility of groups

Recall a result due to O. J. Schmidt that a finite group whose proper subgroups are nilpotent is soluble. The present note extends this result and shows that if all non-normal maximal subgroups of a finite group are nilpotent, then (i) it is soluble; (ii) it is p-nilpotent for some prime p; (iii) if it is not nilpotent, then the number of prime divisors contained in its order is between 2 and k + 2, where k is the number of normal maximal subgroups which are not nilpotent.     

<Emphasis Type="Italic">X</Emphasis>-decomposable finite groups for <Emphasis Type="Italic">X</Emphasis>={1, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis> + 1, <Emphasis Type="Italic">m</Emphasis> + 2}

A normal subgroup N of a finite group G is called n-decomposable in G if N is the union of n distinct G-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is m-decomposable, m+1-decomposable, or m+2-decomposable for some positive integer m. Furthermore, we give classification for the soluble case.     

Existence and Number of Maximal Precompact Topologies on Abelian Groups

The lattice PC(G) of precompact group topologies on an Abelian group G is isomorphic with the lattice SG(G*) of subgroups of the algebraic character group (Remus, 1983). Remus used this result to determine the number of precompact [Hausdorff] topologies on Abelian groups. In this paper the same tool is applied to the problems of existence and number of maximal precompact [Hausdorff] topologies on an Abelian group G, i.e. antiatoms in the lattice PC(G). It is shown that PC(G) has antiatoms iff G is not torsion-free. Further the number of maximal precompact [Hausdorff] topologies is expressed in terms of the cardinalities of the p-components of the group G.     

On second maximal subgroups of Sylow subgroups of finite groups

Finite groups in which the second maximal subgroups of the Sylow p-subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.     

On Conjugacy p-Separability of Free Products of Two Groups with Amalgamation

It is proved that a free product of two finite p-groups with amalgamated central subgroups is a conjugacy p-separable group. With the help of this result, it is proved that a free product with amalgamated subgroups of two finitely generated Abelian groups is a residually finite p-group if and only if it is conjugacy p-separable.     

Factorisation Theorems for T-decomposable Measures on Groups

 For a real or p-adic unipotent algebraic group G, given a T∈ Hom(G, G) and T-decomposable measure on G which is either ‘full’ or symmetric, we get a decomposition , where μ₀ is T-invariant and , and this decomposition is unique upto a shift. We also show that ν₀ is T-decomposable under some additional sufficient condition and give a counter example to justify this. We generalise the above to power bounded operators on p-adic Banach spaces. We also prove some convergence-of-types theorems on p-adic groups as well as Banach spaces. (Received 21 October 2000; in revised form 21 February 2001)     

Frattini subgroups and supernilpotent groups

In the context of the problem of which nonabelianp-groups can occur as normal subgroups contained in Frattini subgroups, the family of supernilpotent groups (all maximal subgroups characteristic) is investigated. Results of this investigation are applied to the Frattini-embedding problem, incorporating recent work of A. R. Makan. The groups of order 2ⁿ (n ≦ 6) have been examined with respect to supernilpotence and their occurrence as normal subgroups contained in Frattini subgroups. Results of this examination are presented.     

On the number of maximal subgroups of a finite solvable group

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \mathfrak m_p(G){\mathfrak {m}_{p}(G)} , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.     

Finite non-elementary abelian p-groups whose number of subgroups is maximal

Assume G is a direct product of M _p (1, 1, 1) and an elementary abelian p-group, where M _p (1, 1, 1) = 〈a, b | a ^p = b ^p = c ^p =1, [a,b]=c,[c,a] = [c,b]=1〉. When p is odd, we prove that G is the group whose number of subgroups is maximal except for elementary abelian p-groups. Moreover, the counting formula for the groups is given.     

On K-ℙ-subnormal subgroups of finite groups

A subgroup H of a group G is said to be K-?-subnormal in G if H can be joined to the group by a chain of subgroups each of which is either normal in the next subgroup or of prime index in it. Properties of K-?-subnormal subgroups are obtained. A class of finite groups whose Sylow p-subgroups are K-?-subnormal in G for every p in a given set of primes is studied. Some products of K-?-subnormal subgroups are investigated.