Verifying Huppert’s Conjecture for <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and cd(G) be the set of 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 twisted Ree groups ² G ₂(q ²) for q ² = 3^2m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.     

On capability of finite abelian groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the group of inner automorphisms of some group) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H _i } of subgroups of G with trivial intersection, such that the union generates G and all quotients G/H _i have the same exponent. Other variations of this condition are also provided (for instance, the condition that the union generates G can be replaced by the condition that it is equal to G). The work presented here is partially supported by NSF/DMS-0805932.     

The Isomorphism Problem for Integral Group Rings of a Complete Monomial Group

Let G be a complete monomial group with abelian base, namely, G = AwrSym _m , the wreath product of a finite abelian group A with the symmetric group on m letters. Then the group G is determined by its integral group ring.     

Verifying Huppert's Conjecture for PSL3(q) and PSU3(q 2)

Let G be a finite group and cd(G) be the set of 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. We examine arguments to verify this conjecture for the simple groups of Lie type of rank two. To illustrate our arguments, we extend Huppert's results and verify the conjecture for the simple linear and unitary groups of rank two.     

On P𝔉-Supplemented subgroups of finite groups

Let 𝔉 be a class of groups and G a finite group. A maximal subgroup M of G is called 𝔉-abnormal provided G∕M_G?𝔉. Let K<H be subgroup of G. Then we say that (K,H) is an 𝔉-abnormal pair of G provided K is a maximal 𝔉-abnormal subgroup of H. Let A be a subgroup of G. Then we say that A is 𝔉-quasipermutable in G provided A either covers or avoids every 𝔉-abnormal pair of G. In this paper, we consider some applications of 𝔉-quasipermutable subgroups.     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

On certain homomorphisms of restriction algebras of symmetric sets

LetG be a locally compact abelian group and Γ its dual group. For any closedH ⊆G denote the algebra of restrictions toH of Fourier transforms of functions inL ¹(Γ) byA(H). This paper considers certain Cantor like sets inR and ΠZ _m(j) and gives some necessary algebraic criterion fornatural isomorphisms of their restriction algebras. This work was supported mainly by the U.S. National Science Foundation Graduate Fellowship Program. The author wishes to thank Paul Cohen, Karel de Leeuw, and Yitzhak Katznelson for their counsel.     

Abelian groups with a vanishing homology group

An abelian group A is called absolutely abelian, if in every central extension N ? G ? A the group G is also abelian. The abelian group A is absolutely abelian precisely when the Schur multiplicator H₂A vanished. These groups, and more generally groups with H_nA = 0 for some n, are characterized by elementary internal properties. (Here H₁A denotes the integral homology of A.) The cases of even n and odd n behave strikingly different. There are 2^?ο different isomorphism types of abelian groups A with reduced torsion subgroup satisfying H_2nA = 0. The major tools are direct limit arguments and the Lyndon-Hochschild-Serre (L-H-S) spectral sequence, but the treatment of absolutely abelian groups does not use spectral sequences. All differentials d^r for r ≥ 2 in the L-H-S spectral sequence of a pure abelian extension vanish. Included is a proof of the folklore theorem, that homology of groups commutes with direct limits also in the group variable, and a discussion of the L-H-S spectral sequence for direct limits.     

STACKED BASES FOR HOMOGENEOUS COMPLETELY DECOMPOSABLE GROUPS

Generalizing a theorem by P. Hill and C. Megibben, fixing a rational group R, we characterize by numerical invariants R-presentations of a group G, namely, short exact sequences of the form 0 → A → X → G → 0, where A and X are homogeneous completely decomposable groups of the same type R. This characterization sets afloat the class of the “uniquely R-presented groups”. This class is investigated in connection with the extension to arbitrary groups of the Warfield equivalence between categories of torsionfree abelian groups induced by the functors Hom(R, –) and R ? ?. As an application, the stacked bases theorem proved by J. Cohen and H. Gluck in 1970 is extended to arbitrary pairs of homogeneous completely decomposable abelian groups of the same type.

     

A new characterization of frobenius complements

LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC _G(a) for 1≠a ∈A.     

Verifying Huppert’s Conjecture for PSp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) when <Emphasis Type="Italic">q</Emphasis> > 7

Let G denote a finite group and cd (G) the set of 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. Huppert verified the conjecture for PSp₄(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp₄(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank two.     

Cleavages of graphs: the spectral radius

A cleavage of a finite graph G is a morphism f : H → G of graphs such that if P is the m × n characteristic matrix defined as P _ik = 1 if i ∈ f ^?1(k), otherwise = 0, then A(H)P ≤ PA(G), where A(G) and A(H) are the adjacency matrices of G and H, respectively. This concept generalizes induced subgraphs, quotients of graphs, Galois covers, path-tree graphs and others. We show that for spectral radii we have the inequality ρ(H) ≤ ρ(G). Equality holds only in case f : H → G is an equivariant quotient and H has isoperimetric constant i(H) = 0.     

A Family of Non-quasiprimitive Graphs Constructed From PSU3（q） Where q Is Odd

Let Г be a simple connected graph and let G be a group of automorphisms of Г. Г is said to be （G, 2）-arc transitive if G is transitive on the 2-arcs of Г. It has been shown that there exists a family of non-quasiprimitive （PSU3（q）, 2）-arc transitive graphs where q = 2^3m with m an odd integer. In this paper we investigate the case where q is an odd prime power.     

On automorphism groups of Cayley graphs

LetX _G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(X_G,H) must contain the regular subgroupL _G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(X_G,H)L_G] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.     

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.     

On G-Covering Subgroup Systems for Some Saturated Formations of Finite Groups

Let ? be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for ? if G ∈ ? whenever Σ ? ?. For a non-identity subgroup H of G, we put Σ _H be some set of subgroups of G which contains at least one supplement in G of each maximal subgroup of H. Let p ≠ q be primes dividing |G|, P, and Q be non-identity a p-subgroup and a q-subgroup of G, respectively. We prove that Σ _P and Σ _P  ∪ Σ _Q are G-covering subgroup systems for many classes of finite groups.     

On automorphisms of finite Abelian <Emphasis Type="Italic">p</Emphasis>-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A _p . For a finite abelian p-group A of type (k ₁, ..., k _n ), simple necessary and sufficient conditions for an n × n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k ₁, k ₂) is analyzed. This work has begin during the visit of the second author to the Faculty of Mathematics and Computer Science, Nicolaus Copernicus University during the period July 31–August 13, 2005. This visit was supported by the Nicolaus Copernicus University and a grant from Cnpq.     

A class of Banach algebras whose duals have the Radon-Nikodym property

Let A be a complex, commutative Banach algebra and let M_A be the structure space of A. Assume that there exists a continuous homomorphism h : L¹(G) → A with dense range, where L¹(G) is the group algebra of a locally compact abelian group G. The main results of this paper can be summarized as follows: (a) If the dual space A* has the Radon-Nikodym property, then M_A is scattered (i.e., it has no nonempty perfect subset) and . (b) If the algebra A has an identity, then the space A* has the Radon-Nikodym property if and only if . Furthermore, any of these conditions implies that M_A is scattered. Several applications are given. Received: 29 September 2005     

Simplicity of Skew Group Rings of Abelian Groups

Necessary and sufficient conditions for simplicity of a general skew group ring A ?_σ G are not known. In this article, we show that a skew group ring A ?_σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ?_σ G is simple.