Characterizations of schunck classes of finite groups

We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f,g∈R[X] (the ideal of R generated by the coefficients of f g is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a _i b _j=0 for each coefficient a _i of f and b _j of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regularR is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.     

On complemented subgroups of finite groups

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H ⋂ K = 1. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about p-nilpotent groups.     

On complemented subgroups of finite groups

In this paper, it is proved that the class of all finite supersoluble groups with elementary abelian Sylow subgroups is just the class of all finite groups for which every minimal subgroup is complemented. The structure of a finite group under the assumption that all maximal subgroups (respectively 2-maximal) of any Sylow subgroup are complemented is also analyzed.     

On two classes of finite supersoluble groups

Let ? be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ?-S-semipermutable if H permutes with every Sylow p-subgroup of G in ? for all p?π(H); H is said to be ?-S-seminormal if it is normalized by every Sylow p-subgroup of G in ? for all p?π(H). The main aim of this paper is to characterize the ?-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ? are ?-S-semipermutable in G and the ?-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ? are ?-S-seminormal in G.     

Conjugacy classes in finite groups

In the first part of this note, we give new proofs of known results regarding the class number of finite groups, adding a few related results. In the second part, we improve a result of Ito concerning a special class ofp-groups.     

On the number of conjugacy classes of finite nilpotent groups

We establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group. In particular, we obtain that for any constant c there are only finitely many finite p-groups of order pm with at most c⋅m conjugacy classes. This answers a question of L. Pyber.     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

Approximability of finite rank soluble groups by certain classes of finite groups

For soluble groups of finite rank we obtain the necessary and sufficient condition to be a virtually residually finite p-group. We also prove that a soluble group G of finite rank is residually π-finite for some finite set π of primes if and only if it has no subgroups of type Q and the torsion radical of G is a finite group.     

Conjugacy classes in finite solvable groups

In this note, we obtain the number of conjugacy classes in a finite solvable group as a function of any tuple of the composition factors ofG. Using this relation, we give a new elementary proof of one of Mann's results for solvable groups, without using character theory, and we improve this result for some classes of groups.     

On a graph associated to invariant conjugacy classes of finite groups

LetA andG be finite groups of coprime orders such thatA acts by automorphisms onG. We define theA-invariant conjugacy class graph ofG to be the graph having as vertices the noncentralA-invariant conjugacy classes ofG, and two vertices are connected by an edge if their cardinalities are not coprime. We prove that when the graph is disconnected thenG is solvable.     

Products of fitting classes of finite groups

Using Fitting classes we generalize some well known theorems on centralizers in finite groups.     

Equivalence classes of functions between finite groups

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals and splitting relative difference sets (RDSs) and introduce an intermediate relation, canonical equivalence, to aid in distinguishing the classes. We identify very precisely the conditions under which a graph equivalence determines a bundle equivalence, using transversals and extensions. We derive a new and easily computed algebraic measure of nonlinearity for a function f, calculated from the image of its coboundary ∂f. This measure is preserved by bundle equivalence but not by the coarser equivalences. It takes its minimum value if f is a homomorphism, and takes its maximum value if the graph of f contains a splitting RDS.     

Some finite groups with large conjugacy classes

We derive some properties of a family of finite groups, which was investigated by Camina, Macdonald, and others. For instance, we give information about the Schur multipliers of the class twop-groups in this family. A large part of this paper was written while the author was visiting the Department of Mathematics of the University of Trento. The author is indebted to this department, and in particular to C.M. Scoppola, for their kind hospitality. The author is also grateful to D. Chillag for his constructively destructive criticism of the first version of this paper.