On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

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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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Integral group rings of solvable groups with trivial central units

Forum Mathematicum ◽

10.1515/forum-2017-0021 ◽

2018 ◽

Vol 30 (4) ◽

pp. 845-855 ◽

Cited By ~ 5

Author(s):

Andreas Bächle

Keyword(s):

Solvable Group ◽

Group Ring ◽

Solvable Groups ◽

Integral Group Ring ◽

Finite Solvable Group ◽

Group Rings ◽

Integral Group ◽

Frobenius Groups ◽

Integral Group Rings ◽

Rational Groups

Abstract The integral group ring {\mathbb{Z}G} of a group G has only trivial central units if the only central units of {\mathbb{Z}G} are {\pm z} for z in the center of G. We show that the order of a finite solvable group G with this property can only be divisible by the primes 2, 3, 5 and 7, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.

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Units and augmentation powers in integral group rings

Journal of Group Theory ◽

10.1515/jgth-2020-0050 ◽

2020 ◽

Vol 23 (6) ◽

pp. 931-944

Author(s):

Sugandha Maheshwary ◽

Inder Bir S. Passi

Keyword(s):

Present Article ◽

Group Ring ◽

Unit Group ◽

Integral Group Ring ◽

Group Rings ◽

Integral Group ◽

Integral Group Rings ◽

Natural Filtration

AbstractThe augmentation powers in an integral group ring {\mathbb{Z}G} induce a natural filtration of the unit group of {\mathbb{Z}G} analogous to the filtration of the group G given by its dimension series {\{D_{n}(G)\}_{n\geq 1}}. The purpose of the present article is to investigate this filtration, in particular, the triviality of its intersection.

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On Class Sums in p-Adic Group Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-058-5 ◽

1971 ◽

Vol 23 (3) ◽

pp. 541-543 ◽

Cited By ~ 5

Author(s):

Sudarshan K. Sehgal

Keyword(s):

Number Field ◽

Group Ring ◽

Group Algebra ◽

Nilpotent Groups ◽

Nilpotency Class ◽

Group Rings ◽

Integral Group ◽

Group Automorphism ◽

Integral Group Rings

In this note we prove that an isomorphism of p-adic group rings of finite p-groups maps class sums onto class sums. For integral group rings this is a well known theorem of Glauberman (see [3; 7]). As an application, we show that any automorphism of the p-adic group ring of a finite p-group of nilpotency class 2 is composed of a group automorphism and a conjugation by a suitable element of the p-adic group algebra. This was proved for integral group rings of finite nilpotent groups of class 2 in [5]. In general this question remains open. We also indicate an extension of a theorem of Passman and Whitcomb. The following notation is used.G denotes a finite p-group.Z denotes the ring of (rational) integers.ZP denotes the ring of p-adic integers.Qp denotes the p-adic number field.

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Units in Integral Group Rings for Order pq

Canadian Journal of Mathematics ◽

10.4153/cjm-1995-006-4 ◽

1995 ◽

Vol 47 (1) ◽

pp. 113-131

Author(s):

Klaus Hoechsmann

Keyword(s):

Abelian Group ◽

Group Ring ◽

Finite Index ◽

Finite Abelian Group ◽

Integral Group Ring ◽

Group Rings ◽

Integral Group ◽

Integral Group Rings ◽

Group Of Units ◽

Cyclotomic Units

AbstractFor any finite abelian group A, let Ω(A) denote the group of units in the integral group ring which are mapped to cyclotomic units by every character of A. It always contains a subgroup Y(A), of finite index, for which a basis can be systematically exhibited. For A of order pq, where p and q are odd primes, we derive estimates for the index [Ω(A) : Y(A)]. In particular, we obtain conditions for its triviality.

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A Remark on the Group Rings of Order Preserving Permutation Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-1968-081-9 ◽

1968 ◽

Vol 11 (5) ◽

pp. 679-680 ◽

Cited By ~ 7

Author(s):

R.H. LaGrange ◽

A.H. Rhemtulla

Keyword(s):

Ordered Set ◽

Isomorphism Problem ◽

Affirmative Answer ◽

Permutation Groups ◽

Group Rings ◽

Integral Group ◽

Integral Group Rings ◽

Totally Ordered Set

If G and H are two groups such that their integral group rings Z(G) and Z(H) are isomorphic, does it follow that G and H are isomorphic? This is the isomorphism problem and an affirmative answer is obtained in case G is a sub group of the group of order preserving permutations of a totally ordered set.

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On the Gruenberg–Kegel graph of integral group rings of finite groups

International Journal of Algebra and Computation ◽

10.1142/s0218196717500308 ◽

2017 ◽

Vol 27 (06) ◽

pp. 619-631 ◽

Cited By ~ 6

Author(s):

W. Kimmerle ◽

A. Konovalov

Keyword(s):

Group Ring ◽

Finite Groups ◽

Prime Graph ◽

Unit Group ◽

Integral Group Ring ◽

Simple Groups ◽

Group Rings ◽

Integral Group ◽

Almost Simple Groups ◽

Integral Group Rings

The prime graph question asks whether the Gruenberg–Kegel graph of an integral group ring [Formula: see text], i.e. the prime graph of the normalized unit group of [Formula: see text], coincides with that one of the group [Formula: see text]. In this note, we prove for finite groups [Formula: see text] a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups [Formula: see text] whose order is divisible by at most three primes and show that the Gruenberg–Kegel graph of such groups coincides with the prime graph of [Formula: see text].

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On the Structure of Integral Group Rings of Sporadic Groups

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157000000309 ◽

2000 ◽

Vol 3 ◽

pp. 274-306 ◽

Cited By ~ 6

Author(s):

Frauke M. Bleher ◽

Wolfgang Kimmerle

Keyword(s):

Automorphism Groups ◽

Symmetric Groups ◽

Computational Procedure ◽

Isomorphism Problem ◽

Simple Groups ◽

Group Rings ◽

Integral Group ◽

Sporadic Groups ◽

Integral Group Rings ◽

Sporadic Simple Groups

AbstractThe object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

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