# Zassenhaus conjecture

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 Subsequently, they also gave a counterexample to ZC2) (unpublished), which appears in a modified form in [a5].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

How to Cite This Entry:
Zassenhaus conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zassenhaus_conjecture&oldid=51508
This article was adapted from an original article by S.K. Sehgal (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article