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     

Finite solvable groups in which the Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups are either bicyclic or of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript>

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p ³ for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A₄-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p ² + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.     

A construction of two distinct canonical sets of lifts of Brauer characters of a p-solvable group

In [5], Navarro defines the set , where Q is a p-subgroup of a p-solvable group G, and shows that if δ is the trivial character of Q, then Irr(G|Q, δ) provides a set of canonical lifts of IBr_p(G), the irreducible Brauer characters with vertex Q. Previously, in [2], Isaacs defined a canonical set of lifts B_π(G) of I_π(G). Both of these results extend the Fong-Swan Theorem to π-separable groups, and both construct canonical sets of lifts of the generalized Brauer characters. It is known that in the case that 2∈π, or if |G| is odd, we have B_π(G) = Irr(G|Q, 1_Q). In this note we give a counterexample to show that this is not the case when . It is known that if and χ∈B_π(G), then the constituents of χ_N are in B_π (N). However, we use the same counterexample to show that if , and χ∈Irr(G|Q, 1_Q) is such that θ ∈Irr(N) and [θ, χ _N] ≠ 0, then it is not necessarily the case that θ ∈Irr(N) inherits this property. Received: 17 October 2005     

On Two Theorems of Finite Solvable Groups

For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.     

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.     

On a relation between the Sylow and Baer-Suzuki theorems

Given a set π of primes, say that a finite group G satisfies the Sylow π-theorem if every two maximal π-subgroups of G are conjugate; equivalently, the full analog of the Sylow theorem holds for π-subgroups. Say also that a finite group G satisfies the Baer-Suzuki π-theorem if every conjugacy class of G every pair of whose elements generate a π-subgroup itself generates a π-subgroup. In this article we prove, using the classification of finite simple groups, that if a finite group satisfies the Sylow π-theorem then it satisfies the Baer-Suzuki π-theorem as well.     

On the Modular Version of the Gluck-Wolf Theorem

Let N be a normal subgroup of a finite group G. Let ϕ be an irreducible Brauer character of N. Assume π is a set of primes and χ(1)/ϕ(1) is a π′-number of any χ∈IBr _p (G/ϕ). If p∤|G:N|, and N is p-solvable, then G/N has an abelian-by-metabelian Hall-π subgroup; If p∉π then G/N has a metabelian Hall-π subgroup. Received February 22, 2000, Accepted May 9, 2001     

Fields of values of Fong characters

If G is a p-solvable finite group with a p-complement H and φ ∈ IBr(G), then P. Fong showed that there exists α ∈ Irr(H) such that α^G=Φ_φ. In this note we prove that α can be chosen such that the field of values index divides φ (1)p. Received: 6 May 2005     

Finite groups with seminormal Sylow subgroups

In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.     

On Baer-Suzuki <Emphasis Type="Italic">π</Emphasis>-theorems

Given a set π of primes, say that the Baer-Suzuki π-theorem holds for a finite group G if only an element of O _π(G) can, together with each conjugate element, generate a π-subgroup. We find a sufficient condition for the Baer-Suzuki π-theorem to hold for a finite group in terms of nonabelian composition factors. We show also that in case 2 ∉ π the Baer-Suzuki π-theorem holds for every finite group.     

On a group with three supersolvable subgroups of pairwise relatively prime indices

In this paper, we show that if G is a finite group with three supersolvable subgroups of pairwise relatively prime indices in G and G′ is nilpotent, then G is supersolvable. Let π(G) denote the set of prime divisors of |G| and max(π(G)) denote the largest prime divisor of |G|. We also establish that if G is a finite group such that G has three supersolvable subgroups H, K, and L whose indices in G are pairwise relatively prime, q \nmid p-1{q \nmid p-1} where p =  max(π(G)) and q = max(π(L)) with L a Hall p′-subgroup of G, then G is supersolvable.     

OD-Characterization of alternating and symmetric groups of degrees 16 and 22

Let G be a finite group and π(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p ₁, p ₂,...,p _k }, where p ₁ < p ₂ < ... < p _k , then the degree pattern of G is defined by D(G) = (deg(p ₁), deg(p ₂),...,deg(p _k )). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |H| = |G| and D(H) = D(G). In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A ₂₂ is OD-characterizable. We also show that the automorphism groups of the alternating groups A ₁₆ and A ₂₂, i.e., the symmetric groups S ₁₆ and S ₂₂ are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.     

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.     

A Remark on З-Permutability of Finite Groups

Let З be a complete set of Sylow subgroups of a finite group G, that is, З contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup of a finite group G is said to be З-permutable if it permutes with every member of З. Recently, using the Classification of Finite Simple Groups, Heliel, Li and Li proved tile following result： If the cyclic subgroups of prime order or order 4 iif p = 2） of every member of З are З-permutable subgroups in G, then G is supersolvable. In this paper, we give an elementary proof of this theorem and generalize it in terms of formation.     

Conjugacy classes and finite p-groups

Let G be a finite p-group, where p is a prime number, and a ∈ G. Denote by Cl(a) = {gag⁻¹| g ∈ G} the conjugacy class of a in G. Assume that |Cl(a)| = pⁿ. Then Cl(a) Cl(a⁻¹) = {xy | x ∈ Cl(a), y ∈ Cl(a⁻¹)} is the union of at least n(p − 1) + 1 distinct conjugacy classes of G. Received: 16 December 2004     

Recognition of some symmetric groups by the set of the order of their elements

For a finite group G, let π_e(G) be the set of order of elements in G and denote S _n the symmetric group on n letters. We will show that if π_e(G ) = π_e(H), where H is S _p or S _p+1 and p is a prime with 50 < p < 100, then G ≅ H. This revised version was published online in August 2006 with corrections to the Cover Date.     

Automorfismi spezzanti di ordine primo

LetG be a group andα εAut(G);α is calledn-splitting if gg^α...g^{α n-1}=1 ∀g εG. In this note we study the structure of finite groups admitting an-splitting automorphism of orderp (p an odd prime number).

     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

Σ-Uniform structures and Σ-functions. II

We construct a family of Σ-uniform Abelian groups and a family of Σ-uniform rings. Conditions are specified that are necessary and sufficient for a universal Σ-function to exist in a hereditarily finite admissible set over structures in these families. It is proved that there is a set S of primes such that no universal Σ-function exists in hereditarily finite admissible sets \mathbbH\mathbbF(G) \mathbb{H}\mathbb{F}(G) and \mathbbH\mathbbF(K) \mathbb{H}\mathbb{F}(K) , where G = ⊕{Z _p | p ∈ S} is a group, Z _p is a cyclic group of order p, K = ⊕{F _p | p ∈ S} is a ring, and F _p is a prime field of characteristic p.