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 Subsequently, they also gave a counterexample to ZC2) (unpublished), which appears in a modified form in [a5].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

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