Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some 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).

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Torsion units in integral group rings of nilpotent metabelian groups

Communications in Algebra ◽

10.1080/00927879208824348 ◽

1992 ◽

Vol 20 (2) ◽

pp. 423-435 ◽

Cited By ~ 5

Author(s):

Michael A. Dokuchaev

Keyword(s):

Group Rings ◽

Integral Group ◽

Metabelian Groups ◽

Integral Group Rings ◽

Torsion Units

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On the Torsion Units of Some Integral Group Rings

Algebra Colloquium ◽

10.1142/s1005386706000290 ◽

2006 ◽

Vol 13 (02) ◽

pp. 329-348 ◽

Cited By ~ 34

Author(s):

Martin Hertweck

Keyword(s):

Finite Group ◽

Finite Simple Group ◽

Fundamental Result ◽

Group Rings ◽

Integral Group ◽

Integral Group Rings ◽

Torsion Unit ◽

Zassenhaus Conjecture ◽

A Finite Group ◽

Torsion Units

It is shown that any torsion unit of the integral group ring ℤG of a finite group G is rationally conjugate to a trivial unit if G = P ⋊ A with P a normal Sylow p-subgroup of G and A an abelian p′-group (thus confirming a conjecture of Zassenhaus for this particular class of groups). The proof is an application of a fundamental result of Weiss. It is also shown that the Zassenhaus conjecture holds for PSL(2,7), the finite simple group of order 168.

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Rational Conjugacy of Torsion Units in Integral Group Rings of Non-Solvable Groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091516000535 ◽

2017 ◽

Vol 60 (4) ◽

pp. 813-830 ◽

Cited By ~ 11

Author(s):

Andreas Bächle ◽

Leo Margolis

Keyword(s):

Prime Graph ◽

Affirmative Answer ◽

New Method ◽

Group Rings ◽

Integral Group ◽

Modular Representation Theory ◽

Integral Group Rings ◽

Zassenhaus Conjecture ◽

Normalized Units ◽

Torsion Units

AbstractWe 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.

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