View source for Zassenhaus conjecture

Just as the only roots of unity in a [[Cyclotomic field|cyclotomic field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300701.png" /> are of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300702.png" />, there is the classical theorem of G. Higman stating that the torsion units in the integral group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300703.png" /> of a finite [[Abelian group|Abelian group]] are of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300705.png" />. Of course, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300706.png" /> is non-Abelian, then any conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300707.png" /> is also of finite order; however, these are not all the torsion units in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300708.png" />. The famous Zassenhaus conjecture says that for a [[Finite group|finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z1300709.png" /> all torsion units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007010.png" /> are rationally conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007012.png" />:

ZC1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007014.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007015.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007016.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007017.png" /> and some unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007018.png" />. This conjecture was proved to be true by A. Weiss, first for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007019.png" />-groups [[#References|[a16]]] and then for nilpotent groups [[#References|[a17]]] (cf. also [[Nilpotent group|Nilpotent group]]). In fact, Weiss proved the following stronger Zassenhaus conjecture for nilpotent groups:

ZC3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007020.png" /> is a finite subgroup of units of augmentation one in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007021.png" />, then there exists a unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007023.png" />. A special case of this is the following conjecture:

ZC2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007024.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007025.png" /> of augmentation one of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007027.png" />, then there exists a unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007029.png" />. This last conjecture was earlier proved by K. Roggenkamp and L.R. Scott [[#References|[a12]]] for nilpotent groups. Subsequently, they also gave a counterexample to ZC2) (unpublished), which appears in a modified form in [[#References|[a5]]].

Clearly, ZC3) implies ZC1) and ZC2). Also, ZC2) implies that if two group rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007031.png" /> are isomorphic, then the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007033.png" /> are isomorphic. This isomorphism problem was proposed in [[#References|[a3]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

Of course, then, (a1) is true for nilpotent groups. Moreover, it was proved by A. Whitcomb [[#References|[a18]]] that (a1) is true for metabelian groups. M. Hertweck [[#References|[a2]]] has given a counterexample to (a1).

Conjecture ZC1) is open in general (as of 2000). Besides nilpotent groups, it is known to be true for certain split metacyclic groups [[#References|[a10]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007035.png" /> is the semi-direct product of two cyclic groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007037.png" /> of relative prime orders, then ZC1) holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007038.png" />.

This result has been strengthened to ZC3) [[#References|[a15]]].

There are several useful and interesting extensions of the above conjectures. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007039.png" /> is a [[Normal subgroup|normal subgroup]] of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007041.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007042.png" /> can be represented by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007043.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007044.png" />. Any torsion unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007046.png" /> that is mapped by the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007047.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007048.png" /> gives rise to a torsion matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007049.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007050.png" /> denotes the subgroup of the [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007051.png" /> consisting of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007052.png" /> that are mapped by the augmentation homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007053.png" />, when applied to each entry, to the identity matrix. Thus, ZC1) translates to the question about diagonalization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007055.png" />: Is a torsion matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007057.png" /> is a finite group, conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007058.png" /> to a matrix of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007060.png" />?

This was answered positively in [[#References|[a16]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007061.png" />-groups (cf. also [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007062.png" />-group]]). See [[#References|[a1]]] for an explicit example of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007063.png" /> that cannot be diagonalized but for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007064.png" />. Such a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007065.png" /> exists for a finite nilpotent group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007066.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007067.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007068.png" /> has at least two non-cyclic Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007069.png" />-subgroups [[#References|[a1]]] (cf. also [[Sylow subgroup|Sylow subgroup]]).

However, it was proved in [[#References|[a6]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007071.png" /> is finite Abelian, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007072.png" /> is conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007073.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007074.png" />. This has been extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007075.png" /> in [[#References|[a8]]], bridging the gap between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007077.png" />.

The Zassenhaus conjectures and the isomorphism problem have also been studied for infinite groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007078.png" />. The statements remain the same and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007079.png" /> is arbitrary. A counterexample to ZC1) was provided in [[#References|[a9]]]. Conjecture ZC2) also does not hold for infinite groups, as shown by S.K. Sehgal and A.E. Zalesskii (see [[#References|[a14]]], p. 279).

However, one can ask if any torsion unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007080.png" /> can be stably diagonalized to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007082.png" />. This has been proved [[#References|[a7]]] to be true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007083.png" />-elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007084.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007085.png" /> is nilpotent. The isomorphism problem also has a positive answer for finitely-generated nilpotent groups of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007086.png" />, cf. [[#References|[a11]]]. In general for nilpotent groups the problem remains open (as of 2000).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Cliff,   A. Weiss,   "Finite groups of matrices over group rings"  ''Trans. Amer. Math. Soc.'' , '''352'''  (2000)  pp. 457–475</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hertweck,   "A solution of the isomorphism problem for integral group rings"  (to appear)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Higman,   "Units in group rings" , ''D. Phil. Thesis Univ. Oxford''  (1940)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Higman,   "The units of group rings"  ''Proc. London Math. Soc.'' , '''46'''  (1940)  pp. 231–248</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Klingler,   "Construction of a counterexample to a conjecture of Zassenhaus"  ''Commun. Algebra'' , '''19'''  (1991)  pp. 2303–2330</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.S. Luthar,   I.B.S. Passi,   "Torsion units in matrix group rings"  ''Commun. Algebra'' , '''20'''  (1992)  pp. 1223–1228</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Z. Marciniak,   S.K. Sehgal,   "Finite matrix groups over nilpotent group rings"  ''J. Algebra'' , '''181'''  (1996)  pp. 565–583</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  Z. Marciniak,   S.K. Sehgal,   "Torsion matrices over abelian group rings"  ''J. Group Th.'' , '''3'''  (2000)  pp. 67–75</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  Z. Marciniak,   S.K. Sehgal,   "Zassenhaus conjecture and infinite nilpotent groups"  ''J. Algebra'' , '''184'''  (1996)  pp. 207–212</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C. Polcino Milies,   J. Ritter,   S.K. Sehgal,   "On a conjecture of Zassenhaus on torsion units in integral group rings, II"  ''Proc. Amer. Math. Soc.'' , '''97'''  (1986)  pp. 201–206</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J. Ritter,   S.K. Sehgal,   "Isomorphism of group rings"  ''Archiv Math.'' , '''40'''  (1983)  pp. 32–39</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  K. Roggenkamp,   L. Scott,   "Isomorphisms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007087.png" />-adic group rings"  ''Ann. Math.'' , '''126'''  (1987)  pp. 593–647</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  S.K. Sehgal,   "Topics in group rings" , M. Dekker  (1978)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  S.K. Sehgal,   "Units in integral group rings" , Longman  (1993)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  A. Valenti,   "Torsion units in integral group rings"  ''Proc. Amer. Math. Soc.'' , '''120'''  (1994)  pp. 1–4</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  A. Weiss,   "Rigidity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007088.png" />-adic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007089.png" />-torsion"  ''Ann. of Math.'' , '''127'''  (1988)  pp. 317–332</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  A. Weiss,   "Torsion units in integral group rings"  ''J. Reine Angew. Math.'' , '''415'''  (1991)  pp. 175–187</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  A. Whitcomb,   "The group ring problem"  ''PhD Thesis Univ. Chicago''  (1968)</TD></TR></table>