Uniquely factorizable elements and solvability of finite groups

In a finite group G every element can be factorized in such a way that there is one factor for each prime divisor p of | G |, and the order of this factor is p^α for some integer α ≧ 0. We define g ∈G to be uniquely factorizable if it has just one such factorization (whose factors must be pairwise commuting). We consider the existence of uniquely factorizable elements and its relation to the solvability of the group. We prove that G is solvable if and only if the set of all uniquely factorizable elements of G is the Fitting subgroup of G. We also prove various sufficient conditions for the non-existence of uniquely factorizable elements in non-solvable groups. Received: 9 June 2005     

On chief factors of finite groups

Let H and K be normal subgroups of a finite group G and let K≤H. If A is a subgroup of G such that AH=AK or A∩H=A∩K, we say that A covers or avoids H/K respectively. The purpose of this paper is to investigate factor groups of a finite group G using this concept. We get some characterizations of a finite group being solvable or supersolvable and generalize some known results.     

On <Emphasis Type="Italic">c</Emphasis>-normal maximal and minimal subgroups of Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups of finite groups

A subgroup H of a finite group G is called c-normal in G if there exists a normal subgroup N of G such that G = HN and $H \cap N \leq H_{G} = {\rm core}_{G}(H)$. In this paper, we investigate the class of groups of which every maximal subgroup of its Sylow p-subgroup is c-normal and the class of groups of which some minimal subgroups of its Sylow p-subgroup is c-normal for some prime number p. Some interesting results are obtained and consequently, many known results related to p-nilpotent groups and p-supersolvable groups are generalized.     

Sylow products and the solvable radical

We study products of Sylow subgroups of a finite group G. First we prove that G is solvable if and only if G = P₁ ... P_m for any choice of Sylow p_i-subgroups P_i , where p₁,..., p_m are all of the distinct prime divisors of |G|, and for any ordering of the p_i . Then, for a general finite group G, we show that the intersection of all Sylow products as above is a subgroup of G which is closely related to the solvable radical of G. Received: 18 November 2004     

The influence of $\pi$-quasinormality of some subgroups of a finite group

A subgroup H of G is said to be $\pi$-quasinormal in G if it permute with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of G are $\pi$-quasinormal in G. The main result we proved in this paper is the following:Theorem 3.4. Let ${\cal F}$ be a saturated formation containing the supersolvable groups. Suppose that G is a group with a normal subgroup H such that $G/H \in {\cal F}$, and all maximal subgroups of any Sylow subgroup of $F^{*}(H)$ are $\pi$-quasinormal in G, then $G \in {\cal F}$. Received: 10 May 2002     

The influence of minimal subgroups of focal subgroups on the structure of finite groups

A subgroup H of a finite 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, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in N_G(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained.     

Characterization of Finite Groups With Some <Emphasis Type="Italic">S</Emphasis>-quasinormal Subgroups

A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.     

A description of Baer-Suzuki type of the solvable radical of a finite group

We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g∈G such that for any 3 elements a₁,a₂,a₃∈G the subgroup generated by the elements , i=1,2,3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2,3}^′-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).     

Products of Subgroups Which Are Subgroups

We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group.     

Groups with a positive law on commutators

Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]ⁱ, a,b,c,d∈G and i?0, then the entire derived subgroup G^′ satisfies a positive law. In fact, G^′ is an extension of a nilpotent group by a locally finite group of finite exponent.     

The automorphism group of a nonsplit metacyclic p-group

This note considers a finite group G = HK, which is a product of a subgroup H and a normal subgroup K, and determines subgroups of Aut G. The special case when G is a nonsplit metacyclic p-group, where p is odd, is then considered and the structure of its automorphism group Aut G is given. Received: 13 September 2007, Revised: 22 November 2007     

Vertices of simple modules and normal subgroups ofp-solvable groups

Let π be a set of prime numbers andG a finite π-separable group. Let θ be an irreducible π′-partial character of a normal subgroupN ofG and denote by I_π′ (G‖θ), the set of all irreducible π′-partial characters Φ ofG such that θ is a constituent of Φ_N. In this paper, we obtain some information about the vertices of the elements in I_π′ (G‖θ). As a consequence, we establish an analogue of Fong's theorem on defect groups of covering blocks, for the vertices of the simple modules (in characteristicsp) of a finitep-solvable group lying over a fixed simple module of a normal subgroup.     

A characterization of the hypercyclically embedded subgroups of finite groups

A normal subgroup H of a finite group G is said to be hypercyclically embedded in G if every chief factor of G below H is cyclic. The major aim of the present paper is to characterize the normal hypercyclically embedded subgroups E of a group G by means of the embedding of the maximal and minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of E.     

Derived length and conjugacy class sizes

Let G be a finite solvable group, and let F(G) be its Fitting subgroup. We prove that there is a universal bound for the derived length of G/F(G) in terms of the number of distinct conjugacy class sizes of G. This result is asymptotically best possible. It is based on the following result on orbit sizes in finite linear group actions: If G is a finite solvable group and V a finite faithful irreducible G-module of characteristic r, then there is a universal logarithmic bound for the derived length of G in terms of the number of distinct r^′-parts of the orbit sizes of G on V. This is a refinement of the author's previous work on orbit sizes.     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.     

A probabilistic version of a theorem of Kegel

Let G be a finite group and α an automorphism of G of prime order p. It is shown that if the elements x of G satisfy the equation with sufficiently high probability (depending only on p) then G is nilpotent.Received: 8 January 2005     

Totally Permutable Products of Finite Groups Satisfying SC or PST

Finite groups G=AB factorized by two subgroups A and B such that every subgroup of A permutes with every subgroup of B are studied in this paper. The behaviour of such products with respect to the class of finite groups in which Sylow-permutability is transitive is analyzed.     

On orbits of automorphism groups II

Let G be a finite group, let A be a group of automorphisms of G and let C_G(A) denote the subgroup of fixed points of A in G. If the order of C_G(A) is coprime to the number of orbits of A in G, then C_G(A) is contained in the autocommutator subgroup [G, A]. The notion of class-avoiding automorphism is used to extend theorems of J. Thompson and P. Rowley. Received: 3 November 2008, Revised: 1 December 2008     

A note on subgroup commutativity degrees of finite groups

In this note we give some new results concerning the subgroup commutativity degree of a finite group G. These are obtained by considering the minimum of subgroup commutativity degrees of all sections of G.