Finite groups with C-quasinormal subgroups

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

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.

Variants of some of the Brauer-Fowler theorems

A subgroup was defined by O. Ore to be quasinormal in a group if it permuted with all subgroups of the group, and he proved [5] that such a subgroup is subnormal (= subinvariant = accessible) in a finite group. Finite groups in which all subgroups are quasinormal were classified by K. Iwasawa [3], and more recently N. Ito and J. Szép [2] and the author [1] proved that a quasi-normal subgroup is an extension of a normal subgroup by a nilpotent group. Similar results were obtained by O. Kegel [4] and in [1] for subgroups which permute not necessarily with all subgroups but with those having some special property.

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.

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogue remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups A_5 and A_6, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal 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-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 &lt; |D| &lt; |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

Finite Groups of Chain Difference One

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic biprimitive finite groups saturated with groups of the sets {L2(pn)}, {Re(32n+1)}, and {Sz(22n+1)}. It is proved thai such groups are all isomorphic to {L2(P)}, {Re(Q)}, or {Sr(Q)} over locally finite fields.

We show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G-module V such that V∣H is free. The problem is raised in the recent paper by Kuzucuŏglu and Zalesskiǐ [15] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n, there exists a polynomial GLn(Q)-module V of dimension 2 n Π p p ( 2 n ) , where the product is over all primes less than or equal to n+1, such that V is free as a QH-module for every finite subgroup H of GLn(Q). The module V is the tensor product of the exterior algebra Λ*(E), on the natural GLn(Q)-module E, and Steinberg modules Stp, one for each prime not exceeding n+1. The Steinberg modules also play the major role in the case in which K has characteristic p>0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K, the Steinberg module St p n for G is injective (and projective) on restriction to H for n≫0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π GL(V)→GL(W) such that, for each faithful representation ρ: H→GL(V), the composite πορ: H→GL(W) is free, in particular all representations πορ are equivalent.

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

Suppose that \(G\) is a finite group and \(H\) is a subgroup of \(G\). \(H\) is said to be an \(s\)-quasinormally embedded in \(G\) if for each prime \(p\) dividing the order of \(H\), a Sylow \(p\)-subgroup of \(H\) is also a Sylow \(p\)-subgroup of some \(S\)-quasinormal subgroup of \(G\); \(H\) is said to be \(c\)-normal in \(G\) if \(G\) has a normal subgroup \(T\) such that \(G=HT\) and \(H\cap T\le H_{G}\), where \(H_{G}\) is the normal core of \(H\) in \(G\). We fix in every non-cyclic Sylow subgroup \(P\) of \(G\) some subgroup \(D\) satisfying \(1<|D|<|P|\) and study the structure of \(G\) under the assumption that every subgroup \(H\) of \(P\) with \(|H|=|D|\) is either \(s\)-quasinormally embedded or \(c\)-normal in \(G\). Some recent results are generalized and unified.

In this paper we classify finite simple groups whose Sylow $2$-subgroups have nilpotency class two.

LetG be a finite group. We say thatG satisfiesI(k) if anyk+1 distinct Sylow 2-subgroups ofG interesect trivially. In this paper we classify all finite groups satisfyingI (3), and all finite simple groups satisfyingI (5).

Introduction. In recent years, there have been interests in studying the influence of arithmetic conditions on the lengths of conjugacy classes to the structure of finite groups. The results are parallel to those of character degrees, and lead to strong restrictions to the group structure. An interesting relevant topic is the lengths of conjugacy classes of subgroups. This was first brought up to the author by Professor J. P. Zhang, who has obtained some interesting results on conjugacy class lengths of Sylow subgroups, the so called Sylow numbers [91. In this paper, we consider conjugacy class lengths of cyclic subgroups of finite goups. Our notations are standard and tends to follow [5]. All groups considered are finite. Let G be a finite group, H a subgroup of G, then the length of conjugacy class of H, the number of subgroups of G conjugate to H, is equal to I G : NG(H)[. Our main result is Theorem 1, its proof depends on the classification of finite simple groups.