Embedding free products in the unit group of an integral group ring

We study the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M24. As a consequence, for this group we give a positive answer to the question by Kimmerle about prime graphs.

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INVOLUTIONS AND FREE PAIRS OF BASS CYCLIC UNITS IN INTEGRAL GROUP RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498811004872 ◽

2011 ◽

Vol 10 (04) ◽

pp. 711-725 ◽

Cited By ~ 2

Author(s):

J. Z. GONÇALVES ◽

D. S. PASSMAN

Keyword(s):

Group Ring ◽

Unit Group ◽

Integral Group Ring ◽

Group Rings ◽

Integral Group ◽

Nonabelian Group ◽

Ring Of Integers ◽

Integral Group Rings ◽

Noncyclic Subgroup

Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk, m(x), uk, m(x*)) or (uk, m(x), uk, m(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.

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The Hypercentre and the n-Centre of the Unit Group of an Integral Group Ring

Canadian Journal of Mathematics ◽

10.4153/cjm-1998-021-2 ◽

1998 ◽

Vol 50 (2) ◽

pp. 401-411 ◽

Cited By ~ 14

Author(s):

Yuanlin Li

Keyword(s):

Group Ring ◽

Periodic Group ◽

Unit Group ◽

Complete Characterization ◽

Integral Group Ring ◽

Integral Group

AbstractIn this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most 2. We then give a complete characterization of the n-centre of that unit group. The n-centre of the unit group is either the centre or the second centre (for n ≥ 2).

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The Upper Central Series of the Unit Group of an Integral Group Ring

Communications in Algebra ◽

10.1081/agb-200061026 ◽

2005 ◽

Vol 33 (5) ◽

pp. 1409-1415 ◽

Cited By ~ 3

Author(s):

Yuanlin Li ◽

M.M. Parmenter

Keyword(s):

Group Ring ◽

Unit Group ◽

Integral Group Ring ◽

Central Series ◽

Integral Group

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Units of group rings of groups of order 16

Glasgow Mathematical Journal ◽

10.1017/s0017089500009952 ◽

1993 ◽

Vol 35 (3) ◽

pp. 367-379 ◽

Cited By ~ 8

Author(s):

E. Jespers ◽

M. M. Parmenter

Keyword(s):

Finite Group ◽

Group Ring ◽

Abelian Groups ◽

Unit Group ◽

Integral Group Ring ◽

Group Rings ◽

Integral Group ◽

Group Of Units ◽

Abelian Case ◽

A Finite Group

LetGbe a finite group,(ZG) the group of units of the integral group ring ZGand1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively(ZG) for particular groupsG.This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described(ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of(ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

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