Partial augmentations power property: A Zassenhaus Conjecture related problem

Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group G is conjugate in the rational group algebra of G to an element in ±G. We review the known weaker versions of this conjecture and introduce a new condition, on the partial augmentations of the powers of a unit of finite order in $\mathbb{Z}G$, which is weaker than the Zassenhaus Conjecture but stronger than its other weaker versions.We prove that this condition is satisfied for units mapping to the identity modulo a nilpotent normal subgroup of G. Moreover, we show that if the condition holds then the HeLP Method adopts a more friendly form and use this to prove the Zassenhaus Conjecture for a special class of groups.     

Traces of Torsion Units

A conjecture due to Zassenhaus asserts that if G is a finite group then any torsion unit in ?G is conjugate in ?G to an element of G. Here, a weaker form of this conjecture is proved for some infinite groups.     

Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some metabelian groups

We prove a conjecture of Zassenhaus that every normalized torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a group element for some metabelian groups including metacyclic groups G containing a normal cyclic group A such that G/A is cyclic of prime power order. The relative prime case was done in [11]. Received: 21 April 2005     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

Zassenhaus猜想的一个注记

在这篇注记中,通过研究两个有限群直积的整群环的正规化挠单位的偏增广,证明了在某些条件下这些正规化挠单位与该群中的元素在有理群代数中共轭.     

Group Gradings in Incidence Algebras

If X is a bounded countable locally finite partially ordered set, R an integral domain, and G a group having the property that no non-identity element has order a unit of R. Then it is shown that any G-grading of the incidence algebra I(X, R) is equivalent to a good grading. Further, an example is given showing that not all group gradings of incidence algebras are equivalent to good gradings.     

On the order of the unitary subgroup of a modular group algebra

Let KG be a group algebra of a finite p-group G over a finite field Kof characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A.     

Transitive partial actions of groups

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y _G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y _G such that the global group action of G on Y _G is continuous and the injection Y into Y _G is an open dense equivariant embedding.      

A Note on the Infimum of Energy of Unit Vector Fields on a Compact Riemannian Manifold

Our main result in this paper establishes that if G is a compact Lie subgroup of the isometry group of a compact Riemannian manifold M acting with cohomogeneity one in M and either G has no singular orbits or the singular orbits of G have dimension at most n−3, then the unit vector field N orthogonal to the principal orbits of G is weakly smooth and is a critical point of the energy functional acting on the unit normal vector fields of M. A formula for the energy of N in terms of the of integral of the Ricci curvature of M and of the integral of the square of the mean curvature of the principal orbits of G is obtained as well. In the case that M is the sphere and G the orthogonal group it is known that that N is minimizer. It is an open question if N is a minimizer in general.     

Estimating L ∞ Norms by L 2k Norms for Functions on Orbits

Abstract. Let G be a compact group acting in a real vector space V . We obtain a number of inequalities relating the L ^∞ norm of a matrix element of the representation of G with its L ^2k norm for a positive integer k . As an application, we obtain approximation algorithms to find the maximum absolute value of a given multivariate polynomial over the unit sphere (in which case G is the orthogonal group) and for the assignment problem of degree d , a hard problem of combinatorial optimization generalizing the quadratic assignment problem (in which case G is the symmetric group).     

Groups whose vanishing class sizes are not divisible by a given prime

Let G be a finite group. An element ${g\in G}Let G be a finite group. An element g ? G{g\in G} is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.     

Invariants of an augmentation of Γ0(n)

Let G _n ^× be the 2-group of primary factors of a positive integer n and fix a direct product decomposition of this group. We define an augmentation of Γ₀(n) based on G _n ^×, paralleling augmentations used by Fricke, Cohn and Knopp, and others. Using the decomposition of G _n ^×, we then define a family of functions based on η-functions and use these functions to construct invariants of the augmented group. Along with proving results analogous to those of Cohn and Knopp, we make a complete determination of the multiplier systems for these new functions.     

Torsion units in integral group rings of metacyclic groups

It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of $\text{Z}$G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (|A|, |X|) = 1 then any torsion unit of $\text{Z}$G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.     

On Groups with the Second Largest Value of the Sum of Element Orders

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group ? _n . In this paper, we investigate groups with the second largest value of ψ on the set of groups of the same order.     

On the analogue of Veech’s theorem in the WAP-compactification of a locally compact group

Let G be a non-compact locally compact group with an identity e and let UG be the LUC -compactification of G . Veech's theorem asserts that gx and x are distinct in UG whenever g is an element in G other than e. We study the analogue of this theorem in the WAP -compactification of G.     

Bases for permutation groups and matroids

In this paper, we give two equivalent conditions for the irredundant bases of a permutation group to be the bases of a matroid. (These are deduced from a more general result for families of sets.) If they hold, then the group acts geometrically on the matroid, in the sense that the fixed points of any element form a flat. Some partial results towards a classification of such permutation groups are given. Further, if G acts geometrically on a perfect matroid design, there is a formula for the number of G-orbits on bases in terms of the cardinalities of flats and the numbers of G-orbits on tuples. This reduces, in a particular case, to the inversion formula for Stirling numbers.     

On a problem from the Kourovka Notebook

In this article, it is proved that if a group G coincides with its commutator subgroup, is generated by a finite set of classes of conjugate elements, and contains a proper minimal normal subgroup A such that the factor group G/A coincides with the normal closure of one element, then G coincides with the normal closure of an element. From this a positive answer to question 5.52 from the Kourovka Notebook for the group with the condition of minimality on normal subgroups follows. We have found a necessary and sufficient condition for a group coinciding with its commutator subgroup and generated by a finite set of classes of conjugate elements not to coincide with the normal closure of any element. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 119–125, 2005.