Finite p-groups with a minimal non-abelian subgroup of index p(Ⅱ)

We classify completely three-generator finite p-groups G such that Ф(G)≤Z(G)and|G′|≤p2.This paper is a part of the classification of finite p-groups with a minimal non-abelian subgroup of index p,and solve partly a problem proposed by Berkovich.     

Finite groups with some pronormal subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H ^x in 〈H, H ^x 〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.     

On maximal abelian subgroups in finite 2-groups

We determine here up to isomorphism the structure of any finite nonabelian 2-group G in which every two distinct maximal abelian subgroups have cyclic intersection. We obtain five infinite classes of such 2-groups (Theorem 1.1). This solves for p = 2 the problem Nr. 521 stated by Berkovich (in preparation). The more general problem Nr. 258 stated by Berkovich (in preparation) about the structure of finite nonabelian p-groups G such that A ∩ B = Z(G) for every two distinct maximal abelian subgroups A and B is treated in Theorems 3.1 and 3.2. In Corollary 3.3 we get a new result for an arbitrary finite 2-group. As an application of Theorems 3.1 and 3.2, we solve for p = 2 a problem of Heineken-Mann (Problem Nr. 169 stated in Berkovich, in preparation), classifying finite 2-groups G such that A/Z(G) is cyclic for each maximal abelian subgroup A (Theorem 4.1).      

Maximal subrings and <Emphasis Type="Italic">E</Emphasis>-groups

For a finite group G, let E(G) denote the near-ring of functions generated by the semigroup, End(G), of endomorphisms of G. We characterize when E(G) is maximal as a subnear-ring of M ₀(G). A group G is an E-group if E(G) is a ring. We give a new characterization of finite E-groups and investigate the problem of determining, for finite E-groups, when E(G) is maximal as a ring in M₀(G). Received: 26 June 2006     

Finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions

We classify here nonmetacyclic finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions (Theorem 1.1). This solves problem Nr. 710 for p = 2 stated by Y. Berkovich in [1]. For p > 2 the corresponding problem is open. Received: 14 March 2007 Revised: 15 April 2007     

On finite <Emphasis Type="Italic">p</Emphasis>-groups whose automorphisms are all central

An automorphism α of a group G is said to be central if α commutes with every inner automorphism of G. We construct a family of non-special finite p-groups having abelian automorphism groups. These groups provide counterexamples to a conjecture of A. Mahalanobis [Israel J. Math. 165 (2008), 161–187]. We also construct a family of finite p-groups having non-abelian automorphism groups and all automorphisms central. This solves a problem of I. Malinowska [Advances in Group Theory, Aracne Editrice, Rome, 2002, pp. 111–127].     

ON MINIMAL NON-<Emphasis Type="Italic">MSP</Emphasis>-GROUPS

A finite group G is called an MSP-group if all maximal subgroups of the Sylow subgroups of G are S-quasinormal in G: We give a complete classification of groups that are not MSP-groups but all their proper subgroups are MSP-groups.     

On <Emphasis Type="Italic">T</Emphasis>-groups,supersolvable groups,and maximal subgroups

A group is called a T-group if all its subnormal subgroups are normal. Finite T-groups have been widely studied since the seminal paper of Gaschütz (J. Reine Angew. Math. 198 (1957), 87–92), in which he described the structure of finite solvable T-groups. We call a finite group G an NNM-group if each non-normal subgroup of G is contained in a non-normal maximal subgroup of G. Let G be a finite group. Using the concept of NNM-groups, we give a necessary and sufficient condition for G to be a solvable T-group (Theorem 1), and sufficient conditions for G to be supersolvable (Theorems 5, 7 and Corollary 6).     

Finite Soluble Groups with Permutable Subnormal Subgroups

A finite group G is said to be a PST-group if every subnormal subgroup of G permutes with every Sylow subgroup of G. We shall discuss the normal structure of soluble PST-groups, mainly defining a local version of this concept. A deep study of the local structure turns out to be crucial for obtaining information about the global property. Moreover, a new approach to soluble PT-groups, i.e., soluble groups in which permutability is a transitive relation, follows naturally from our vision of PST-groups. Our techniques and results provide a unified point of view for T-groups, PT-groups, and PST-groups in the soluble universe, showing that the difference between these classes is quite simply their Sylow structure.     

On equality of central and class preserving automorphisms of finite p-groups

Let G be a finite non-abelian p-group, where p is a prime. Let Aut_c(G) and Aut_z(G) respectively denote the group of all class preserving and central automorphisms of G. We give a necessary and sufficient condition for G such that Aut_c(G) = Aut_z(G) and classify all finite non-abelian p-groups G with elementary abelian or cyclic center such that Aut_c(G) = Aut_z(G). We also characterize all finite p-groups G of order ≤ p ⁷ such that Aut_z(G) = Aut_z(G) and complete the classification of all finite p-groups of order ≤ p ⁵ for which there exist non-inner class preserving automorphisms.     

Finite 2-groups all of whose nonabelian subgroups are generated with involutions

In this note we determine finite nonabelian 2-groups G all of whose nonabelian subgroups are generated by involutions and show that such groups must be quasi-dihedral. This solves the problem Nr. 1595 for p = 2 in [1].     

Finite p-Groups All of Whose Minimal Nonnormal Subgroups Posses Large Normal Closures

We obtained some results about finite p-groups G with G/H^G being abelian for all nonnormal subgroups H, where H^G denotes the normal closure of H. Moreover, we give a classification of finite p-groups G with G/H^G being cyclic for all nonnormal subgroups H.     

Finiteness of Higher K -Groups of Orders and Group Rings

In this paper, we prove that if R is the ring of integers in a number field F, A any R-order in a semisimple F-algebra, then K_2n(A), G_2n(A) are finite groups for all positive integers n. Hence, even dimensional higher K- and G-groups of integral grouprings of finite groups are finite. We also show that in odd dimensions, SK_n of integral and p-adic integral grouprings of finite p-groups are also finite p-groups (Received: August 2005)     

On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

Finite p-groups not characteristic in any p-group in which they are properly contained

Answering a question raised by Y. Berkovich, we give examples of finite p-groups G with the property that the only finite p-group K with G char K, is G itself. We also prove a theorem stating that every finite p-group is contained in such a group G.     

Finite groups in which permutability is a transitive relation on their frattini factor groups

Let G be a finite group. A PT-group is a group G whose subnormal subgroups are all permutable in G. A PST-group is a group G whose subnormal subgroups are all S-permutable in G. We say that G is a PT_o-group (respectively, a PST_o-group) if its Frattini quotient group G/Φ(G) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PT_o-groups and minimal non-PST_o-groups.      

Finite groups with many product conjugacy classes

We classify all finite groupsG such that the product of any two non-inverse conjugacy classes ofG is always a conjugacy class ofG. We also classify all finite groupsG for which the product of any twoG-conjugacy classes which are not inverse modulo the center ofG is again a conjugacy class ofG.     

子群的核平凡或正规闭包极大的有限p群

称有限 p 群 G 为ACT 群,如果对每个交换子群H, 其正规核 H_G=1 或 H_G=H. 又称p 群 G是CC 群,如果对每个非正规交换子群H, 有 H_G=1 或 H^G 在G中的指数为 p. 本文分类了ACT 群和CC 群.     

A Classification of Minimal Non-MNP-Groups

A finite group G is called an MNP-group if all maximal subgroups of every Sylow subgroup of G are normal in G. In this article, we give a complete classification of those groups which are not MNP-groups but all of whose proper subgroups are MNP-groups.     

C55-Groups

We classify the C55-groups, i.e., finite groups in which the centralizer of every 5-element is a 5-group.