Existence of Hall normal subgroups in finite groups

We obtain a necessary and sufficient criterion for the existence of an invariant complement to a nilpotent subgroup contained as a direct factor in one of the maximal subgroups of a given group; we also find a condition for the p-closure of a group, all proper subgroups of which are p-closed, expressed in terms of the degree of one of its nonlinear irreducible characters.Translated from Matematicheskie Zametki, Vol. 16, No. 3, pp. 381–385, September, 1974.     

p-groups without noncharacteristic normal subgroups

Let P be a finite p-group, p a prime. We prove that there is a finite p-group Q ≥ P such that every normal subgroup of Q is characteristic in Q.     

The enumeration of subgroups in finite p-groups

We generalize the familiar principle of enumeration due to Hall and establish a new principle for the enumeration of subgroups of any p-group G of order p^m, based on the following grouptheoretic relation found by the author: $$\sum\nolimits_{\lambda = 0}^m {\left( { - 1} \right)^\lambda p^{\left( {\begin{array}{*{20}c} \lambda \\ 2 \\ \end{array} } \right)} \mathcal{E}_\lambda \left( G \right)} = 0,$$ where ?_λ (G) is the number of elementary Abelian subgroups of order p^λ in G.     

On maximal cyclic subgroups in finite p-groups

In this note we consider finite noncyclic p-groups G all of whose maximal cyclic subgroups X satisfy one of the following two properties. (a) If each subgroup H of G containing X properly is nonabelian, then p = 2 and G is generalized quaternion. (b) If X is contained in exactly one maximal subgroup of G, then G is metacyclic. This solves the problems Nr.1541 and Nr. 1594 from [1].     

Conjugacy classes of non-normal subgroups of finite p-groups

Let G be a finite p-group, and let ν(G) denote the number of conjugacy classes of non-normal subgroups of G. It is known that either ν(G) ≤ 1 or ν(G) ≥ p. We determine all p-groups G with ν(G) ≤ p + 1.     

Representations of elementary abelian p-groups and finite subgroups of fields

Suppose $\mathbb{F}$ is a field of prime characteristic p and E is a finite subgroup of the additive group $(\mathbb{F},+)$. Then E is an elementary abelian p-group. We consider two such subgroups, say E and $E^{\prime }$, to be equivalent if there is an $\alpha \in {\mathbb{F}}^{\times }:=\mathbb{F}?\{0\}$ such that $E=\alpha E^{\prime }$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.     

关于有限p-群的正规闭包的一些条件

Berkovich 提出了研究满足下列条件的有限p-群G, 对于G的每一个非正规子群H满足(N₁)exp(H)=exp(H^G); (N₂) |H^G:H|≤ p; (N₃) H^G=HG''. 本文首先研究满足条件(N₃) 的有限p-群,然后讨论满足条件(N₁); (N₂) 和(N₃) 的有限p-群.     

Pronormality of Hall subgroups in finite simple groups

We prove that the Hall subgroups of finite simple groups are pronormal. Thus we obtain an affirmative answer to Problem 17.45(a) of the Kourovka Notebook.     

On the poset of conjugacy classes of subgroups of finite p-groups

This paper is a continuation of the paper “On the Deskins completions, theta completions and theta pairs for maximal subgroups ”.In the former paper, Zhao Yaoqing introduced the concept of θ-completions associated to a maximal subgroup of a finite group. The concept offers a convenience for us to study the completions introduced by Deskins and gives us a way to reveal the relationship between the concepts of completions and θ-pairs, the latter concept is introduced by Mukherjee and Bhattacharya. The present paper is devoted to discussing the π-solvability, π-supersolvability and π-nilpotency of a finite group by using the θ-completions. Moreover, a new proof on the Deskins conjecture concerning the supersolvability is included.     

On Hall subgroups of a finite group

New criteria of existence and conjugacy of Hall subgroups of finite groups are given.     

Posets of subgroups in p-groups

It is shown that certain classes of finite p-groups are characterized by the ranked partially ordered set of conjugacy classes of their subgroups.     

The lower bound of the number of nonabelian subgroups of finite p-groups

In this paper, we give the lower bound of the number of nonabelian subgroups of possible index of finite p-groups, and classify the finite p-groups such that the number of nonabelian subgroups of possible order are exactly the lower bound.     

The number of generators and orders of Abelian subgroups of finite p-groups

Let f(F) be the smallest function such that every finite p-group, all of whose Abelian subgroups are generated by at most n elements (all of whose Abelian subgroups have orders at most p_n, has at most f(n) generators (has order not exceeding p^F(n)). It is established that the functions f and F have quadratic order of growth.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 337–341, March, 1978.     

Fong characters and chains of normal subgroups

In this note, we show that if is a π-partial character of the π-separable group is a chain of normal subgroups of G, and H is a Hall π-subgroup of G, then has a Fong character α Irr(H) such that for every subgroup , every irreducible constituent of α _H∩N is Fong for N. We also show that if is quasi-primitive, then for every normal subgroup M of G the irreducible constituents of are Fong for M. Received: 21 July 2006 Revised: 17 January 2007     

A conjugacy criterion for Hall subgroups in finite groups

A finite group G is said to satisfy C _π for a set of primes π, if G possesses exactly one class of conjugate π-Hall subgroups. We obtain a criterion for a finite group G to satisfy C _π in terms of a normal series of the group.