Torsion units of integral group rings of metacyclic groups

A weaker version of the Zassenhaus conjecture for torsion units in integral group rings ZG is proved if G is either abelian-by-polycyclic or metabelian. As a consequence we obtain Bovdi's conjecture for torsion units in ZG for metabelian 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).

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.

Let G = ⟨ a ⟩ ⋊ X G = \left \langle a \right \rangle \rtimes X where ⟨ a ⟩ \left \langle a \right \rangle is a cyclic group of order n , X n,X is an abelian group of order m m , and ( n , m ) = 1 (n,m) = 1 . We prove that if Z G \mathbb {Z}G is the integral group ring of G G and H H is a finite group of units of augmentation one of Z G \mathbb {Z}G , then there exists a rational unit γ \gamma such that H γ ⊆ G {H^\gamma } \subseteq G .

This paper is intended to give a survey of recent work on central units in integral group rings. For units in general, the definitive reference is the book by Sehgal (1993) while a survey paper by Jespers contains additional very recent results. Both of these sources contain results on central units (in fact, Jespers devotes a chapter to the topic), but our work complements theirs in two ways. Firstly, we describe some results contained in papers which were not available to the other authors. Secondly, we choose to emphasize some topics which are mentioned either very briefly or not at all in their work. Nevertheless, we acknowledge that there is considerable overlap, especially between our survey and that of Jespers, and would like to thank him for supplying us with a preprint.

(1993). Jordan decomposition and hypercentral units in integral group rings. Communications in Algebra: Vol. 21, No. 1, pp. 25-35.

In this study, we construct a new public key cryptosystem both like El-Gamal and RSA cryptosystem based on Bass cyclic units in integral group rings. Naturally, the underlying hard problems for this system are both decisional Diffie-Hellman assumption and factorization of integers. Since the system is related to RSA scheme indirectly, key generation process is very efficient. This cryptosystem actually supports and strengthens the current public key cryptosystems in [6], [8] and [20]. Encryption algorithm in the system is constructed via non-commutative operations in integral group rings of dihedral groups. Finally, we briefly discuss the security of our cryptosystem against some known attacks.

Torsion units in integral group rings of some metabelian groups, II

Suppose that a group G has a normal subgroup C where C and G/C are cyclic of relatively prime orders.

Several special cases of the conjectures of Bovdi and Zassenhaus are proved. We also deal with special cases of the following conjecture: let α be a torsion unit of the integral group ring ZZG and m the smallest positive integer such that αm ∈G then, m is a divisor of the exponent of the quotient group G/Z(G) provided this exponent is finite

Suppose that a group G G has a normal subgroup C C where C C and G / C G/C are cyclic of relatively prime orders. Then any torsion unit in Z G ZG is rationally conjugate to a trivial unit.

Torsion units in integral group rings of some metabelian groups

Special cases of Bovdi's conjecture are proved.In particular the conjecture is proved for supersolvable and Frobenius groups.We also prove that if exp(G/Z>) is finite, a e VZG a torsion unit and m the smallest positive integer such that a" 1 G G then m divides exp(G/ Z).Let G be a group and let VZG be the group of units of augmentation one of the integral group ring ZG.Given an element x = J2x(g)g G ÏG we set= l(mod/?)and T^\x) = 0(modp) for j < n.In particular there is an element g G G such that o{x) = o(g).Considering these statements he conjectured that if JC is as in Lemma 1 then BC1: T^\x) = 1 and T^\x) = 0 for/ < n.In [4] BC1 is proved for metabelian nilpotent groups and in [2] it is proved in general for nilpotent groups.Bovdi also conjectured the following [1]: BC2: Letw = exp(G/Z(G)) be finite, where Z(G) denotes the center of G.If a G VZG is a torsion unit and m is the smallest positive integer such that a™ e G, then m divides n.We recall that H. J. Zassenhaus had conjectured the following: ZC1: Let G be a finite group and a G VZG a torsion unit then a is conjugated in QG, to an element of G. Lemma 1.1 below shows that ZC1 implies BC1.In this paper we deal with the conjectures BC1 and BC2 and show that BC1 holds for Frobenius groups and polycyclic groups whose commutator subgroup is nilpotent.In particular we re-obtain the result of [2] that BC1 holds for nilpotent groups.Also, we show that BC2 is true for all groups.In the text we denote by S nj -the Kronecker delta function which is 0 if/ ^ n and 1 if j = n.

Torsion units in integral group rings, conjugacy classes globally versus locally

We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus conjecture for the group PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M10 and PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.