Namespaces
Variants
Actions

Difference between revisions of "Zassenhaus conjecture"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (fix tex)
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
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" />:
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,  
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
  
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:
+
Out of 89 formulas, 86 were replaced by TEX code.-->
  
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:
+
{{TEX|done}}
 +
Just as the only roots of unity in a [[Cyclotomic field|cyclotomic field]] $\mathbf{Q} ( \zeta )$ are of the form $\pm \zeta ^ { i }$, there is the classical theorem of G. Higman stating that the torsion units in the integral group ring ${\bf Z} G$ of a finite [[Abelian group|Abelian group]] are of the form $\pm g$, $g \in G$. Of course, if $G$ is non-Abelian, then any conjugate of $\pm g$ is also of finite order; however, these are not all the torsion units in ${\bf Z} G$. The famous Zassenhaus conjecture says that for a [[Finite group|finite group]] $G$ all torsion units of ${\bf Z} G$ are rationally conjugate to $\pm g$, $g \in G$:
  
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]]].
+
ZC1) Let $u \in {\bf Z} G$, $u ^ { n } = 1$ for some $n$; then $u = \pm x ^ { - 1 } g x$ for some $g \in G$ and some unit $x \in \mathbf{Q} G$. This conjecture was proved to be true by A. Weiss, first for $p$-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:
  
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]]]:
+
ZC3) If $H$ is a finite subgroup of units of augmentation one in ${\bf Z} G$, then there exists a unit $x \in \mathbf{Q} G$ such that $x ^ { - 1 } H x \subseteq G$. A special case of this is the following conjecture:
  
<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>
+
ZC2) If $H$ is a subgroup of ${\bf Z} G$ of augmentation one of order $| G |$ such that ${\bf Z} G = {\bf Z} H$, then there exists a unit $x \in \mathbf{Q} G$ with $x ^ { - 1 } H x = G$. 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 ${\bf Z} G$ and $\mathbf{Z}H$ are isomorphic, then the groups $G$ and $H$ are isomorphic. This isomorphism problem was proposed in [[#References|[a3]]]:
 +
 
 +
\begin{equation} \tag{a1} \mathbf{Z} G \simeq \mathbf{Z} H \Rightarrow G \simeq H. \end{equation}
  
 
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).
 
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" />.
+
Conjecture ZC1) was disproved in 2018 [[#References|[a19]]]. Besides nilpotent groups, it is known to be true for certain split metacyclic groups [[#References|[a10]]]: If $G = \langle a \rangle \rtimes \langle  b  \rangle$ is the semi-direct product of two cyclic groups $\langle a \rangle$ and $( b )$ of relative prime orders, then ZC1) holds for $G$.
  
 
This result has been strengthened to ZC3) [[#References|[a15]]].
 
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" />?
+
There are several useful and interesting extensions of the above conjectures. Suppose that $A$ is a [[Normal subgroup|normal subgroup]] of index $n$ in $G$. Then ${\bf Z} G$ can be represented by $( n \times n )$-matrices over $\mathbf{Z} A$. Any torsion unit $u$ of ${\bf Z} G$ that is mapped by the natural homomorphism $G \rightarrow G / A$ to $1 \in \mathbf{Z }( G / A )$ gives rise to a torsion matrix $U \in \text{SGL} _ { n } ( \mathbf{Z} A )$. Here, $SGL_n( \mathbf{Z} A )$ denotes the subgroup of the [[General linear group|general linear group]] $\operatorname{GL} _ { n } ( {\bf Z} A )$ consisting of the matrices $U$ that are mapped by the augmentation homomorphism $\mathbf{Z} A \rightarrow Z$, when applied to each entry, to the identity matrix. Thus, ZC1) translates to the question about diagonalization of $U$ in $\operatorname{GL} _ { n } ( {\bf Q} A )$: Is a torsion matrix $U \in \operatorname{SGL} _ { n } ( \mathbf{Z} G )$, where $G$ is a finite group, conjugate in $\mathbf{Q}G_{n\times n}$ to a matrix of the form $\operatorname{diag} (g_1, \dots , g _ { n } )$, $g_i \in G$?
  
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]]).
+
This was answered positively in [[#References|[a16]]] for $p$-groups (cf. also [[P-group|$p$-group]]). See [[#References|[a1]]] for an explicit example of a matrix $U \in v\operatorname{SGL} _ { 6 } ( \mathbf Z ( C _ { 6 } \times C _ { 6 } ) )$ that cannot be diagonalized but for which $U ^ { 6 } = I$. Such a matrix $U$ exists for a finite nilpotent group $G$ and some $n$ if and only if $G$ has at least two non-cyclic Sylow $p$-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" />.
+
However, it was proved in [[#References|[a6]]] that if $n = 2$ and $G$ is finite Abelian, then $U$ is conjugate in $\mathbf{Q}G_{2\times 2}$ to $\operatorname{diag} (g_1, g_2 )$. This has been extended to $n \leq 5$ in [[#References|[a8]]], bridging the gap between $2$ and $6$.
  
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).
+
The Zassenhaus conjectures and the isomorphism problem have also been studied for infinite groups $\Gamma$. The statements remain the same and the group $\Gamma$ 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).
+
However, one can ask if any torsion unit $U \in \operatorname{SGL} _ { n } ( \Gamma )$ can be stably diagonalized to $\operatorname { diag } ( \gamma _ { 1 } , \ldots , \gamma _ { n } )$, $\gamma _ { i } \in \Gamma$. This has been proved [[#References|[a7]]] to be true for $p$-elements $U$ when $\Gamma$ is nilpotent. The isomorphism problem also has a positive answer for finitely-generated nilpotent groups of class $2$, cf. [[#References|[a11]]]. In general for nilpotent groups the problem remains open (as of 2000).
  
 
====References====
 
====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>
+
<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 $p$-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 $p$-adic $p$-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><tr><td valign="top">[a19]</td> <td valign="top">  F. Eisele, L. Margolis,  "A counterexample to the first Zassenhaus conjecture"  ''Advances Math.'', ''339'''  (2018)  pp. 599–641</td></tr></table>

Latest revision as of 17:05, 26 January 2021

Just as the only roots of unity in a cyclotomic field $\mathbf{Q} ( \zeta )$ are of the form $\pm \zeta ^ { i }$, there is the classical theorem of G. Higman stating that the torsion units in the integral group ring ${\bf Z} G$ of a finite Abelian group are of the form $\pm g$, $g \in G$. Of course, if $G$ is non-Abelian, then any conjugate of $\pm g$ is also of finite order; however, these are not all the torsion units in ${\bf Z} G$. The famous Zassenhaus conjecture says that for a finite group $G$ all torsion units of ${\bf Z} G$ are rationally conjugate to $\pm g$, $g \in G$:

ZC1) Let $u \in {\bf Z} G$, $u ^ { n } = 1$ for some $n$; then $u = \pm x ^ { - 1 } g x$ for some $g \in G$ and some unit $x \in \mathbf{Q} G$. This conjecture was proved to be true by A. Weiss, first for $p$-groups [a16] and then for nilpotent groups [a17] (cf. also Nilpotent group). In fact, Weiss proved the following stronger Zassenhaus conjecture for nilpotent groups:

ZC3) If $H$ is a finite subgroup of units of augmentation one in ${\bf Z} G$, then there exists a unit $x \in \mathbf{Q} G$ such that $x ^ { - 1 } H x \subseteq G$. A special case of this is the following conjecture:

ZC2) If $H$ is a subgroup of ${\bf Z} G$ of augmentation one of order $| G |$ such that ${\bf Z} G = {\bf Z} H$, then there exists a unit $x \in \mathbf{Q} G$ with $x ^ { - 1 } H x = G$. This last conjecture was earlier proved by K. Roggenkamp and L.R. Scott [a12] for nilpotent groups. Subsequently, they also gave a counterexample to ZC2) (unpublished), which appears in a modified form in [a5].

Clearly, ZC3) implies ZC1) and ZC2). Also, ZC2) implies that if two group rings ${\bf Z} G$ and $\mathbf{Z}H$ are isomorphic, then the groups $G$ and $H$ are isomorphic. This isomorphism problem was proposed in [a3]:

\begin{equation} \tag{a1} \mathbf{Z} G \simeq \mathbf{Z} H \Rightarrow G \simeq H. \end{equation}

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

Conjecture ZC1) was disproved in 2018 [a19]. Besides nilpotent groups, it is known to be true for certain split metacyclic groups [a10]: If $G = \langle a \rangle \rtimes \langle b \rangle$ is the semi-direct product of two cyclic groups $\langle a \rangle$ and $( b )$ of relative prime orders, then ZC1) holds for $G$.

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

There are several useful and interesting extensions of the above conjectures. Suppose that $A$ is a normal subgroup of index $n$ in $G$. Then ${\bf Z} G$ can be represented by $( n \times n )$-matrices over $\mathbf{Z} A$. Any torsion unit $u$ of ${\bf Z} G$ that is mapped by the natural homomorphism $G \rightarrow G / A$ to $1 \in \mathbf{Z }( G / A )$ gives rise to a torsion matrix $U \in \text{SGL} _ { n } ( \mathbf{Z} A )$. Here, $SGL_n( \mathbf{Z} A )$ denotes the subgroup of the general linear group $\operatorname{GL} _ { n } ( {\bf Z} A )$ consisting of the matrices $U$ that are mapped by the augmentation homomorphism $\mathbf{Z} A \rightarrow Z$, when applied to each entry, to the identity matrix. Thus, ZC1) translates to the question about diagonalization of $U$ in $\operatorname{GL} _ { n } ( {\bf Q} A )$: Is a torsion matrix $U \in \operatorname{SGL} _ { n } ( \mathbf{Z} G )$, where $G$ is a finite group, conjugate in $\mathbf{Q}G_{n\times n}$ to a matrix of the form $\operatorname{diag} (g_1, \dots , g _ { n } )$, $g_i \in G$?

This was answered positively in [a16] for $p$-groups (cf. also $p$-group). See [a1] for an explicit example of a matrix $U \in v\operatorname{SGL} _ { 6 } ( \mathbf Z ( C _ { 6 } \times C _ { 6 } ) )$ that cannot be diagonalized but for which $U ^ { 6 } = I$. Such a matrix $U$ exists for a finite nilpotent group $G$ and some $n$ if and only if $G$ has at least two non-cyclic Sylow $p$-subgroups [a1] (cf. also Sylow subgroup).

However, it was proved in [a6] that if $n = 2$ and $G$ is finite Abelian, then $U$ is conjugate in $\mathbf{Q}G_{2\times 2}$ to $\operatorname{diag} (g_1, g_2 )$. This has been extended to $n \leq 5$ in [a8], bridging the gap between $2$ and $6$.

The Zassenhaus conjectures and the isomorphism problem have also been studied for infinite groups $\Gamma$. The statements remain the same and the group $\Gamma$ is arbitrary. A counterexample to ZC1) was provided in [a9]. Conjecture ZC2) also does not hold for infinite groups, as shown by S.K. Sehgal and A.E. Zalesskii (see [a14], p. 279).

However, one can ask if any torsion unit $U \in \operatorname{SGL} _ { n } ( \Gamma )$ can be stably diagonalized to $\operatorname { diag } ( \gamma _ { 1 } , \ldots , \gamma _ { n } )$, $\gamma _ { i } \in \Gamma$. This has been proved [a7] to be true for $p$-elements $U$ when $\Gamma$ is nilpotent. The isomorphism problem also has a positive answer for finitely-generated nilpotent groups of class $2$, cf. [a11]. In general for nilpotent groups the problem remains open (as of 2000).

References

[a1] G. Cliff, A. Weiss, "Finite groups of matrices over group rings" Trans. Amer. Math. Soc. , 352' (2000) pp. 457–475
[a2] M. Hertweck, "A solution of the isomorphism problem for integral group rings" (to appear)
[a3] G. Higman, "Units in group rings" , D. Phil. Thesis Univ. Oxford (1940)
[a4] G. Higman, "The units of group rings" Proc. London Math. Soc. , 46 (1940) pp. 231–248
[a5] L. Klingler, "Construction of a counterexample to a conjecture of Zassenhaus" Commun. Algebra , 19 (1991) pp. 2303–2330
[a6] I.S. Luthar, I.B.S. Passi, "Torsion units in matrix group rings" Commun. Algebra , 20 (1992) pp. 1223–1228
[a7] Z. Marciniak, S.K. Sehgal, "Finite matrix groups over nilpotent group rings" J. Algebra , 181 (1996) pp. 565–583
[a8] Z. Marciniak, S.K. Sehgal, "Torsion matrices over abelian group rings" J. Group Th. , 3 (2000) pp. 67–75
[a9] Z. Marciniak, S.K. Sehgal, "Zassenhaus conjecture and infinite nilpotent groups" J. Algebra , 184 (1996) pp. 207–212
[a10] 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
[a11] J. Ritter, S.K. Sehgal, "Isomorphism of group rings" Archiv Math. , 40 (1983) pp. 32–39
[a12] K. Roggenkamp, L. Scott, "Isomorphisms for $p$-adic group rings" Ann. Math. , 126 (1987) pp. 593–647
[a13] S.K. Sehgal, "Topics in group rings" , M. Dekker (1978)
[a14] S.K. Sehgal, "Units in integral group rings" , Longman (1993)
[a15] A. Valenti, "Torsion units in integral group rings" Proc. Amer. Math. Soc. , 120 (1994) pp. 1–4
[a16] A. Weiss, "Rigidity of $p$-adic $p$-torsion" Ann. of Math. , 127 (1988) pp. 317–332
[a17] A. Weiss, "Torsion units in integral group rings" J. Reine Angew. Math. , 415 (1991) pp. 175–187
[a18] A. Whitcomb, "The group ring problem" PhD Thesis Univ. Chicago (1968)
[a19] F. Eisele, L. Margolis, "A counterexample to the first Zassenhaus conjecture" Advances Math., 339 (2018) pp. 599–641
How to Cite This Entry:
Zassenhaus conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zassenhaus_conjecture&oldid=11419
This article was adapted from an original article by S.K. Sehgal (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article