Zassenhaus conjecture for A 6

For the alternating group A ₆ of degree 6, Zassenhaus’ conjecture about rational conjugacy of torsion units in integral group rings is confirmed. Dedicated to the memory of I. S. Luthar Professor Luthar died at the age of 74 in December 2006.     

Integral Group Ring of the Mathieu Simple Group M 23

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M ₂₃ using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M ₁₁ and M ₁₂. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.     

Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionI ⁿ⁺¹ (F) ∩I ⁿ⁺¹ (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I ³ (F)) of F are identified whenR and S are arbitrary subgroups ofF.     

Free Groups and Subgroups of Finite Index in the Unit Group of an Integral Group Ring

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-Abelian group G for which every nonlinear irreducible complex representation is of degree 2 and with commutator subgroup G′ a central elementary Abelian 2-group.     

Vertex-minimal graphs with dihedral symmetry I

Let $D_{2n}$ denote the dihedral group of order $2n$, where $n\ge 3$. In this article, we build upon the findings of Haggard and McCarthy who, for certain values of $n$, produced a vertex-minimal graph with dihedral symmetry. Specifically, Haggard considered the situation when $\frac{n}{2}$ or $n$ is a prime power, and McCarthy investigated the case when $n$ is not divisible by $2$, $3$ or $5$. In this article, we assume $n$ is not divisible by $4$ and construct a vertex-minimal graph whose automorphism group is isomorphic to $D_{2n}$.     

On Fundamental Group of a Certain Class of Welded Knots

In this paper, a certain class of welded knots K_(2n) is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K_(2n),n ∈ Z~+, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.