FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G):={n ∈ ℕ | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ∇(G) ≅ ∇ (M). Then G ≅ M. In this short article we prove that if G is a finite group with ∇(G) ≅ ∇ (A 10), then G ≅ A 10, where A 10 is the alternating group of degree 10.

A subgroup H of a group G is said to be weakly M -supplemented in G if there exists a subgroup B of G provided that (1) G = HB ,a nd (2) ifH1/HG is a maximal subgroup of H/HG ,t henH1B = BH1 <G . where HG is the largest normal subgroup of G contained in H . In this paper we will prove that: Let F be a saturated formation containing all supersolvable groups and G be a group with a normal subgroup H such that G/H ∈F . If every maximal subgroup of every noncyclic Sylow subgroup of F ∗ (H )h aving no supersolvable supplement in G ,i s weaklyM -supplemented in G ,t henG ∈F .

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.

Throughout this note, G will be a finite group, IrrðGÞ will be the set of irreducible characters of G, cdðGÞ will be the set of character degrees of G, and rðGÞ will be the set of primes that divide degrees in cdðGÞ. The prime vertex degree graph of G, written DðGÞ, is the graph with rðGÞ as its vertex set, and with an edge between p and q if pq divides some degree a A cdðGÞ. An overview of the literature on these graphs can be found in [2]. It seems that most groups G have graphs DðGÞ that are complete graphs (although we do not want to be precise on what we mean by ‘most’). For solvable groups, it has been shown that if DðGÞ is not complete, then the structure of G is limited (see [4] and [6]). In this note, we show that a large class of non-solvable groups have degree graphs that are complete. To study the degree graphs of non-solvable groups, a standard starting point is with the degree graphs of non-abelian simple groups. In a series of papers, Don White has classified the character degree graphs of all non-abelian simple groups. Except for some examples of small rank, nearly all non-abelian simple groups have degree graphs that are complete graphs. These results are summarized in [5]. Extending this idea, this note looks at the characteristically simple groups, groups for which there are exactly two characteristic subgroups. Each such group is a direct product of copies of a fixed simple group. With this in mind, we let G denote G G (n copies); thus S is characteristically simple when S itself is a simple group. We now fix S to be a non-abelian simple group. Because the non-abelian simple groups have already been addressed, we restrict our attention to when n > 1. However, it is relatively easy to show that DðSÞ is complete for n > 1. More generally, the graph DðHÞ associated with any non-abelian group H is always complete for every integer n > 1. Thus we expand our scope and consider extending these groups. We recognize first that S is isomorphic to InnðSÞ. Identifying S with its set of inner automorphisms, our goal herein is to prove the following result.

Let r be a positive rational integer, and G a finite group. A subgroup H of G is called a r minimal (r-maximal) subgroup if there exists a subgroup chain 1 = H o ~< H 1 ~< -.. ~<H, = H(H = Ho ~< Hx ~<'" ~< H, = G), where H i is maximal in Hi+ x, we call Han S-quasinormal subgroup of G, if HP = PH for any Sylow subgroup P of G. If H is S-quasinormal in G, we write H SQN G. A subgroup H of G is called a quasinormal subgroup if HK = KH for any subgroup K of G. We write H QN G when H is a quasinormal subgroup of G. G is said to be an SQN-r-group(QN-r-group, PN-r-group) if every r-minimal subgroup of G is S-quasinormal(quasinormal, normal) in G. Clearly, a PN-r-group is a ON-r-group, and a QN-r-group is an SQN-r-group. Gaschfitz and Ito proved that PNl -g roup G is solvable and its commutator subgroup G' isp-nilpotent for any odd prime p. Hence the Fitting length Fl(G) ~< 3 [5, th5.7 p436]. Buckley proved in [1] that PN-1group of odd order is supersolvable. Sastry and Deskins had studied in I'3] PN-r-group and QNr-group. For instance, they proved that QN-r-group (r = 1,2,3) is solvable and QN-l-group and QN-2-group have Fitting length at most 4, the Fitting length of QN-3-group is not more than 5. This paper will study the SQN-r-groups. In section 2, we explore the properties of S--quasinormal subgroup and prove that the normal Hall subgroups of an S-quasinormal subgroup of a finite group is also an S-quasinormal subgroup. In section 3, we prove that SQN-l-group is solvable and the nitpotent residual is p-nilpotent for any odd prime p, so the Fitting length is at most 3. In section 4, we prove that SQN-2-group and SQN-3-group are either supersolvable or isomorphic to the several classes groups given in section t. Moreover, the commutator subgroups are nilpotent, so the Fitting lengths are at most 2. In section 5, we show that SQN-4-group and SQN-5--group are either solvable and the Fitting lengths at most 3 or non-solvable with only one non-abelian principal factor ~r is isomorphic ~o the simple group PSL(2,p). Groups in this .paper are all of finite order. The author wishes to thank his supervisor Professor Duan Xuefu and-Xu Mingyiao for their encouragement and many instructions.

It is shown in this paper that if  is a class of simple groups such that π() = char , the -saturated formation ℌ generated by a finite group cannot be expressed as the Gaschütz product  ∘  of two non--saturated formations if ℌ ≠ . It answers some open questions on products of formations. The relation between ω-saturated and -saturated formations is also discussed.

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

Suppose that [Formula: see text] is a finite group and [Formula: see text] is a subgroup of [Formula: see text]. [Formula: see text] is said to be an [Formula: see text]-quasinormal subgroup of [Formula: see text] if there is a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] permutes with every Sylow subgroup of [Formula: see text]. In this note, we fix in every non-cyclic Sylow subgroup [Formula: see text] of [Formula: see text] some subgroup [Formula: see text] satisfying [Formula: see text] and study the [Formula: see text]-nilpotency of [Formula: see text] under the assumption that every subgroup [Formula: see text] of [Formula: see text] with [Formula: see text] is [Formula: see text]-quasinormal in [Formula: see text]. The Frobenius theorem is generalized.

Let [Formula: see text] be a saturated formation containing [Formula: see text], and G be a finite group. Li etc. proposed a problem: whether there is a normality such that the following two statements are equivalent: (i) [Formula: see text]. (ii) There exists a normal subgroup H of G such that [Formula: see text] and for each Sylow subgroup P of F*(H), every member in some [Formula: see text] satisfies the above normality in G. In this paper, we find a normality satisfying the above problem. Moreover, by using the concept of ℋ-subgroups, we obtain other results about the influence of the members of some fixed [Formula: see text] on the structure of G.

A subgroup H of a group G is said to be nearly normal in G if it has a finite index in its normal closure H G.T he setnn(G) of nearly normal subgroups of G is a sublattice of the lattice of all subgroups of G. Isomorphisms between lattices of nearly normal subgroups of FC -soluble groups are considered in this paper. In particular, properties of images of normal subgroups under such an isomorphism are investigated. Moreover, it is proved that if G is a supersoluble group and ¯ G is an FC -soluble group such that the lattices nn(G )a ndnn( ¯ G )a re isomorphic, then also ¯ G is supersoluble.

Variants of some of the Brauer-Fowler theorems

On a class of inseparable finite groups

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called $${\mathcal{PT}}$$ -groups. In particular, it is shown that the finite solvable $${\mathcal{PT}}$$ -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not $${\mathcal{PT}}$$ -groups but all subnormal subgroups of defect two are permutable.

On Chevalley-groups acting on projective planes

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.