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Just as the only roots of unity in a [[Cyclotomic field|cyclotomic field]] $\mathbf{Q} ( \zeta )$ are of the form $\pm \zeta ^ { 2 }$, 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 $\pm g$, $g \in G$:

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:

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 [[#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 Subsequently, they also gave a counterexample to ZC2) (unpublished), which appears in a modified form in [[#References|[a5]]].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).

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]]].

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130070/z13007058.png"/> 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 $p$-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 $n = 2$ and $G$ is finite Abelian, then $U$ 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 $n \leq 5$ in [[#References|[a8]]], bridging the gap between $2$ and $6$.

The Zassenhaus conjectures and the isomorphism problem have also been studied for infinite groups $\Gamma$. 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 $U \in \operatorname{SGL} _ { n } ( \Gamma )$ 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====
<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>