Groups of prime power order and their nonabelian tensor squares

We prove that the nonabelian tensor square of a powerful p-group is again a powerful p-group. Furthermore, If G is powerful, then the exponent of G ⊕ G divides the exponent of G. New bounds for the exponent, rank, and order of various homological functors of a given finite p-group are obtained. In particular, we improve the bound for the order of the Schur multiplier of a given finite p-group obtained by Lubotzky and Mann.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

ON NONABELIAN TENSOR PRODUCT MODULO q OF GROUPS

The nonabelian tensor product modulo q of two crossed modules of groups is investigated, where q is a positive integer. It is obtained a six term exact sequence of groups connecting the nonabelian tensor product modulo q with algebraic K-functor K ₂ with Z _q coefficients for (noncommutative) local rings. The notion of q-homology groups of a group G with coefficients in a G-module A is introduced, some its properties and calculations are given. The relationship between q-homology groups and derived functors of tensor product modulo q is studied.     

On nonabelian simple groups having the same prime graph as an alternating group

All instances of coincidence between the prime graphs of nonabelian simple groups G and S are found, where G is an alternating group of degree n ≥ 5 and S is a nonabelian finite simple group. The precise bound of the maximal number of pairwise nonisomorphic nonabelian simple groups with the same prime graph is given in the case that one of these groups is an alternating group.     

Finite 2-groups with a nonabelian Frattini subgroup of order 16

According to a classical result of Burnside, if G is a finite 2-group, then the Frattini subgroup Φ(G) of G cannot be a nonabelian group of order 8. Here we study the next possible case, where G is a finite 2-group and Φ(G) is nonabelian of order 16. We show that in that case Φ(G) ≅ M × C₂, where M ≅ D₈ or M ≅ Q₈ and we shall classify all such groups G (Theorem A). Received: 16 February 2005; revised: 7 March 2005     

About noncommuting graphs

The noncommuting graph ?(G) of a nonabelian finite group G is defined as follows: The vertices of ?(G) are represented by the noncentral elements of G, and two distinct vertices x and y are joined by an edge if xy ≠ yx. In [1], the following was conjectured: Let G and H be two nonabelian finite groups such that ?(G) ? ?(H); then ¦G¦ = ¦H¦. Here we give some counterexamples to this conjecture.     

On topologizable and non-topologizable groups

A group G   is called hereditarily non-topologizable if, for every H?G$H?G$, no quotient of H admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked k-generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.     

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.     

Notes on the length of conjugacy classes of finite groups

The purpose of this paper is to investigate influences of lengths of conjugacy classes of finite groups on the structure of finite groups. We get a necessary and sufficient condition for a finite group G to be equal to O_p(G)×O_p^′(G). We also generalize some results (Comm. Algebra 27 (9) (1999) 4347).     

The largest lengths of conjugacy classes and the Sylow subgroups of finite groups

Let G be a finite nonabelian group, P ∈Syl_p(G), and bcl(G) the largest length of conjugacy classes of G. In this short paper, we prove that in general and |P/O_p(G)| < bcl(G) in the case where P is abelian. Received: 26 December 2004; revised: 26 January 2005     

A characterization of finite symplectic polar spaces of odd prime order

A sufficient condition for the representation group for a nonabelian representation (Definition 1.1) of a finite partial linear space to be a finite p-group is given (Theorem 2.9). We characterize finite symplectic polar spaces of rank r at least two and of odd prime order p as the only finite polar spaces of rank at least two and of prime order admitting nonabelian representations. The representation group of such a polar space is an extraspecial p-group of order p1+2r and of exponent p (Theorems 1.5 and 1.6).     

Strongly connective degree sets II: perfect derived subgroups

When G is a finite nonabelian group, we associate the common-divisor graph with G by letting nontrivial degrees in cd(G) = {χ(1) | χ∈Irr(G)} be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of , and every vertex outside of is adjacent to some member of . When G is nonsolvable, we provide sufficiency conditions for cd(G) to have a strongly connective subset. We also extend a previously known result about groups with nonabelian solvable quotients, and prove for arbitrary groups G that if the associated graph is connected and has a diameter bounded by 2, then indeed cd(G) has a strongly connective subset. The major focus is on when the derived subgroup G′ is perfect. Received: 23 July 2005     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.     

On the finite prime spectrum minimal groups

Let G be a finite group. The set of all prime divisors of the order of G is called the prime spectrum of G and is denoted by π(G). A group G is called prime spectrum minimal if π(G) ≠ π(H) for any proper subgroup H of G. We prove that every prime spectrum minimal group all of whose nonabelian composition factors are isomorphic to the groups from the set {PSL ₂(7), PSL ₂(11), PSL ₅(2)} is generated by two conjugate elements. Thus, we extend the corresponding result for finite groups with Hall maximal subgroups. Moreover, we study the normal structure of a finite prime spectrum minimal group with a nonabelian composition factor whose order is divisible by exactly three different primes.     

Probability that the commutator of two group elements is equal to a given element

In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which |G^′| is prime and G^′≤Z(G) as well as for groups G which |G^′| is prime and G^′∩Z(G)=1. This paper extends results of Rusin [see D.J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1) (1979) 237-247].     

The transvection free groups with polynomial rings of invariants

Let G be a finite linear group containing no transvections. This paper proves that the ring of invariants of G is polynomial if and only if the pointwise stabilizer in G of any subspace is generated by pseudoreflections. Kemper and Malle used the classification of finite irreducible groups generated by pseudoreflections to prove the irreducible case in arbitrary characteristic. We extend their result to the reducible case.     

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).     

On Huppert’s Conjecture for 3 D 4(q), q ≥ 3

Let G be a finite group and cd(G) be the set of all complex irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G???H × A, where A is an abelian group. In this paper, we verify the conjecture for the family of simple exceptional groups of Lie type ³ D ₄(q), when q?≥?3.     

Morphic groups

A group G is called morphic if every endomorphism α:G→G for which Gα is normal in G satisfies G/Gα≅ker(α). This concept originated in a 1976 paper of Gertrude Ehrlich characterizing when the endomorphism ring of a module is unit regular. The concept has been extensively studied in module and ring theory, and this paper investigates the idea in the category of groups. After developing their basic properties, we characterize the morphic groups among the dihedral groups and the groups whose normal subgroups form a finite chain. We investigate when a direct product of morphic groups is again morphic, prove that a finite nilpotent group is morphic if and only if its Sylow subgroups are morphic, and present some results for the case where a p-group is morphic.     

Characters and Sylow 2-subgroups of maximal class revisited

We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.