Torsion Units Research Articles

Torsion units for some untwisted exceptional groups of Lie type

Torsion units for some untwisted exceptional groups of Lie type

Torsion units for some almost simple groups

We investigate the Zassenhaus conjecture regarding rational conjugacy of torsion units in integral group rings for certain automorphism groups of simple groups. Recently, many new restrictions on partial augmentations for torsion units of integral group rings have improved the effectiveness of the Luther-Passi method for verifying the Zassenhaus conjecture for certain groups. We prove that the Zassenhaus conjecture is true for the automorphism group of the simple group PSL(2, 11). Additionally we prove that the Prime graph question is true for the automorphism group of the simple group PSL(2, 13).

Torsion units for a Ree group, Tits group and a Steinberg triality group

We investigate the Zassenhaus conjecture for the Steinberg triality group \({}^3D_4(2^3)\), Tits group \({}^2F_4(2)'\) and the Ree group \({}^2F_4(2)\). Consequently, we prove that the Prime Graph question is true for all three groups.

Torsion units for some projected special linear groups

In this paper, we investigate the Zassenhaus conjecture for $PSL(4,3)$ and $PSL(5,2)$. Consequently, we prove that the Prime graph question is true for both groups.

Torsion units in the integral group ring of PSL(3, 4)

We investigate the Zassenhaus Conjecture for the integral group ring of the simple group PSL(3, 4).

Traces of Torsion Units

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

Revisiting the Zassenhaus conjecture on torsion units for the integral group rings of small groups

In recent years several new restrictions on integral partial augmentations for torsion units of \(\mathbb {Z} G\) have been introduced, which have improved the effectiveness of the Luthar–Passi method for checking the Zassenhaus conjecture for specific groups G. In this note, we report that the Luthar–Passi method with the new restrictions are sufficient to verify the Zassenhaus conjecture with a computer for all groups of order less than 96, except for one group of order 48 – the non-split covering group of S4, and one of order 72 of isomorphism type (C3 × C3) ⋊ D8. To verify the Zassenhaus conjecture for this group we give a new construction of normalized torsion units of \(\mathbb {Q} G\) that are not conjugate to elements of \(\mathbb {Z} G\).

On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes

Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings.

Zassenhaus Conjecture for cyclic-by-abelian groups

Zassenhaus Conjecture for torsion units states that every aug- mentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cyclic-by-abelian groups. In this paper G is a finite group and RG denotes the group ring of G with coefficients in a ringR. The units of RG of augmentation one are usually called normalized units. In the 1960s Hans Zassenhaus established a series of conjectures about the finite subgroups of normalized units of ZG. Namely he conjectured that every finite group of normalized units of ZG is conjugate to a subgroup of G in the units of QG. These conjecture is usually denoted (ZC3), while the version of (ZC3) for the particular case of subgroups of normalized units with the same cardinality as G is usually denoted (ZC2). These conjectures have important consequences. For example, a positive solution of (ZC2) implies a positive solution for the Isomorphism and Automorphism Problems (see (Seh93) for details). The most celebrated positive result for Zassenhaus Conjectures is due to Weiss (Wei91) who proved (ZC3) for nilpotent groups. However Roggenkamp and Scott founded a counterexample to the Automorphism Problem, and henceforth to (ZC2) (see (Rog91) and (Kli91)). Later Hertweck (Her01) provided a counterexample to the Isomorphism Problem. The only conjecture of Zassenhaus that is still up is the version for cyclic sub- groups namely:

TORSION UNITS IN INTEGRAL GROUP RINGS OF CONWAY SIMPLE GROUPS

Using the Luthar–Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups Co1, Co2and Co3.

On the isomorphism problem for the ring of monomial representations of a finite group

In this paper we are concerned with the problem of finding properties of a finite group G in the ring D ( G ) of monomial representations of G. We determine the conductors of the primitive idempotents of Q ( ζ ) ⊗ Z D ( G ) , where ζ ∈ C is a primitive | G | -th root of unity, and prove a structure theorem for the torsion units of D ( G ) . Using these results we show that an abelian group G is uniquely determined by the ring D ( G ) . We state an explicit formula for the primitive idempotents of Z [ ζ ] p ⊗ Z D ( G ) , where Z [ ζ ] p is a localization of Z [ ζ ] . We get further results for nilpotent and p-nilpotent groups and we obtain properties of Sylow subgroups of G from D ( G ) .

Development of a cost-effective torsional unit for rodent long bone assessment

Thorough mechanical testing of rodent bones requires an understanding of bone behavior in a variety of loading modes including tension, compression, bending and shear. While these tests are easily conducted with single axis mechanical testing machines, it may also be desirable to determine torsional properties of bone. Although higher-end materials testing machines will enable torsional and/or rotational testing, simpler, less expensive systems rarely offer these capabilities. In this work, we illustrate the development of a torsional system that uses a simple rack and pinion concept to deliver a rotary motion to bones given the linear motion of a testing machine. As the bone field becomes increasingly interdisciplinary, more biologists and non-test engineers need cost-effective mechanical testing capabilities and the torsional system described here has proven to be more than adequate for standard biomechanical testing requirements. Furthermore, given the small-scale size of rodent long bones, a series of potting/testing fixtures were developed that enabled preparation and handling of the specimens without incurring damage to the bone shafts. Once fabricated the system was used to destructively load mice humeri and femurs and quantify torsional properties.

Torsion units in integral group rings of Janko simple groups

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups J 1 , J 2 and J 3 is the same as that of the normalized unit group of their respective integral group ring.

The Effects of Increased Prosthetic Ankle Motions on the Gait of Persons with Bilateral Transtibial Amputations

To determine whether the provision of prosthetic ankle motion improves walking performance in persons with bilateral transtibial amputations. Crossover experimental design in which 19 persons with bilateral transtibial amputations were fitted with Endolite Multiflex Ankles (flexion unit) and Otto Bock Torsion Adapters (torsion unit) to increase relative motion between the prosthetic foot and socket in the sagittal and transverse planes, respectively. Quantitative gait analyses were performed on subjects as they walked with four prosthetic configurations: baseline without flexion or torsion units, with only the flexion unit, with only the torsion unit, and with both the flexion and torsion units. Data were compared with a control group of 14 able-bodied subjects. The flexion unit increased ankle sagittal plane motion (6-7 degrees) and increased positive ankle power (about 0.17 W/kg). The torsion unit increased transverse plane ankle range of motion by 1-2 degrees. Responses from questionnaires indicated that 14 of the 19 subjects preferred the prosthetic configuration that included both the flexion and torsion units. Further, the subjects perceived that the increased prosthetic ankle motion was particularly beneficial for improving stability while they walked on uneven terrain. Both the subjective and objective results suggest that prosthetic foot and ankle components that allow for greater sagittal and transverse plane rotations provide substantial benefit during walking and should be considered for persons with bilateral transtibial amputations. Nonetheless, clinicians should perform individual and appropriate assessments of patients to ensure that they are capable of using components that may improve mobility while possibly sacrificing some degree of stability.

The Orders of Torsion Units in Integral Group Rings of Finite Solvable Groups

It is shown that for any torsion unit of augmentation one in the integral group ring ℤ G of a finite solvable group G, there is an element of G of the same order.

TORSION UNITS IN INTEGRAL GROUP RINGS OF CERTAIN METABELIAN GROUPS

AbstractIt is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

Zassenhaus conjecture for A 6

For the alternating group A6 of degree 6, Zassenhaus’ conjecture about rational conjugacy of torsion units in integral group rings is confirmed.

Torsion Units in Integral Group Ring of the Mathieu Simple GroupM22

AbstractWe investigate the possible character values of torsion units of the normalized unit group of the integral group ring of the Mathieu sporadic groupM22. We confirm the Kimmerle conjecture on prime graphs for this group and specify the partial augmentations for possible counterexamples to the stronger Zassenhaus conjecture.

Zassenhaus conjecture for central extensions of S 5

We confirm a conjecture of Zassenhaus about rational conjugacy of torsion units in integral group rings for a covering group of the symmetric group S 5 and for the general linear group GL(2, 5). The first result, together with others from the literature, settles the conjugacy question for units of prime-power order in the integral group ring of a finite Frobenius group.

Torsion Units in the Integral Group Ring of the Alternating Group of Degree 6

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring ℤG of a finite group G conjugates to a group element within the rational group algebra ℚG. We investigate the Zassenhaus Conjecture (ZC) and a conjecture by W. Kimmerle about prime graph in the normalized unit group of ℤA6.