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).      

Schur multiplicators of infinite pro-<Emphasis Type="Italic">p</Emphasis>-groups with finite coclass

Let G be an infinite pro-p-group of finite coclass and let M(G) be its Schur multiplicator. For p > 2, we determine the isomorphism type of Hom(M(G), ℤ_p), where ℤ_p denotes the p-adic integers, and show that M(G) is infinite. For p = 2, we investigate the Schur multiplicators of the infinite pro-2-groups of small coclass and show that M(G) can be infinite, finite or even trivial.     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

Infinitely many pairs of cospectral integral regular graphs

A graph G is called integral if all the eigenvalues of the adjacency matrix A(G) of G are integers. In this paper, the graphs G ₄(a, b) and G ₅(a, b) with 2a+6b vertices are defined. We give their characteristic polynomials from matrix theory and prove that the (n+2)-regular graphs G ₄(n, n+2) and G ₅(n, n+2) are a pair of non-isomorphic connected cospectral integral regular graphs for any positive integer n.     

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers Ok of k, which, together with the degree [K:k] of the extension determines the Ok-module structure of OK. We call Rt(k,G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A^′-groups inductively, starting with abelian groups and then considering semidirect products of A^′-groups with abelian groups of relatively prime order and direct products of two A^′-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A^′-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine Rt(k,Dn) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).     

Blocks with nonabelian defect groups which have cyclic subgroups of index p

In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group M _n+1(p) of order p ⁿ⁺¹ and exponent p ⁿ for n \geqslant 2{n \geqslant 2}, and where A is not necessarily a full defect p-block but its defect group P = M _n+1(p) is normal in G. The proof is independent of the classification of finite simple groups.     

Homomorphisms between A-Projective Abelian Groups and Left Kasch-Rings

Glaz and Wickless introduced the class G of mixed abelian groups A which have finite torsion-free rank and satisfy the following three properties: i) A _p is finite for all primes p, ii) A is isomorphic to a pure subgroup of _P A _P and iii) Hom(A, tA) is torsion. A ring R is a left Kasch ring if every proper right ideal of R has a non-zero left annihilator. We characterize the elements A of G such that E(A)/tE(A) is a left Kasch ring, and discuss related results.     

Morava K-Theories of p-Groups with Cyclic Maximal Subgroups and Other Related Groups

Let P be a non-Abelian finite p-group, p odd, with cyclic maximal subgroups, and let K(n)^*(–) denote the nth Morava K-theory at p. In this paper we determine the algebras K(n)^*(BP) and K(n)^*(BG) for all groups G with Sylow p-subgroups isomorphic to P, giving further evidence for the fact that Morava K-theory as an invariant of finite groups, is finer than ordinary modp cohomology. Mathematics Subject Classifications (2000): 55N20, 55N22.     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

The ranks of central unit groups of integral group rings of alternating groups

Let G be a finite group and U(Z(Z G)) be the group of units of the center Z(Z G) of the integral group ring Z G (the central unit group of the ring Z G). The purpose of the present work is to study the ranks r _n of groups U(Z(ZAn)), i.e., of central unit groups of integral group rings of alternating groups A _n . We shall find all values n for r _n = 1 and propose an approach on how to describe the groups U(Z(ZAn)) in these cases, and we will present some results of calculations of r _n for n ≤ 600.     

Restrictions of Irreducible Representations of Classical Algebraic Groups to Root A 1-Subgroups

Abstract

Restrictions of irreducible representations of classical algebraic groups to root A ₁-subgroups, i.e., subgroups of type A ₁ generated by root subgroups associated with two opposite roots, are studied. Composition factors of such restrictions are found in the following cases: for groups of types A _n with n > 2 and D _n , for groups of type B _n , n > 2, and long root subgroups, for groups of type C _n , n > 2, and short root subgroups, and for p-restricted representations of A ₂(K), C ₂(K) (recall that B ₂(K) ? C ₂(K)), and of B _n (K), n > 2, and short root subgroups. Here we assume that p > 2 for G = B _n (K) or C _n (K).     

On one class of modules that are close to Noetherian

We consider an R G-module A over a commutative Noetherian ring R. Let G be a group having infinite section p-rank (or infinite 0-rank) such that C _G (A) = 1, A/C _A (G) is not a Noetherian R-module, but the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank, respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.     

A Note on a Class of Factorized <Emphasis Type="Italic">p</Emphasis>-Groups

In this note we study finite p-groups G = AB admitting a factorization by an Abelian subgroup A and a subgroup B. As a consequence of our results we prove that if B contains an Abelian subgroup of index p ⁿ⁻¹ then G has derived length at most 2n.     

A generalization of Brauer''s map of decomposition

We first reformulate Quillen's localization theorem forG-theory in complicial bi-Waldhausen category setting. Secondly, because of this reformulation, we are able to generalize Brauer's decomposition mapd ₀:G ₀(KG)G ₀(kG) to higherG-theory leveld _n :G _n (KG)G _n (kG),n=0, 1 ..., whereG is a finite group,R a Dedekind domain,m a maximal ideal ofR,K=quotient field ofR andk=R/m.     

On Some Sets of Group Functions

Let G be a group, let A be an Abelian group, and let n be an integer such that n –1. In the paper, the sets _n (G,A) of functions from G into A of degree not greater than n are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from G into A is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if G is finite, then the study of the set _n (G,A) is reduced to that of the set n(G/O ^p (G),A _p ) for primes p dividing G/G. Here O ^p (G) stands for the p-coradical of the group G, A _p for the p-component of A, and G for the commutator subgroup of G.     

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.     

On a Note From Oliver Concerning Generalized Euclidean Group Rings

In an unpublished 1987 letter, Bob Oliver determined which elementary abelian 2-groups have generalized euclidean integral group rings. He produced a filtration of E ₂(R) by normal subgroups, sandwiched between elementary and special linear relative groups, with layers that are second homology groups with mod-2 coefficients. His proof is presented here, with related consequences for some other finite groups.