Zassenhaus conjecture
Just as the only roots of unity in a cyclotomic field are of the form
, there is the classical theorem of G. Higman stating that the torsion units in the integral group ring
of a finite Abelian group are of the form
,
. Of course, if
is non-Abelian, then any conjugate of
is also of finite order; however, these are not all the torsion units in
. The famous Zassenhaus conjecture says that for a finite group
all torsion units of
are rationally conjugate to
,
:
ZC1) Let ,
for some
; then
for some
and some unit
. This conjecture was proved to be true by A. Weiss, first for
-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 is a finite subgroup of units of augmentation one in
, then there exists a unit
such that
. A special case of this is the following conjecture:
ZC2) If is a subgroup of
of augmentation one of order
such that
, then there exists a unit
with
. 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 Subsequently, they also gave a counterexample to ZC2) (unpublished), which appears in a modified form in [a5].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 and
are isomorphic, then the groups
and
are isomorphic. This isomorphism problem was proposed in [a3]:
![]() | (a1) |
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) is open in general (as of 2000). Besides nilpotent groups, it is known to be true for certain split metacyclic groups [a10]: If is the semi-direct product of two cyclic groups
and
of relative prime orders, then ZC1) holds for
.
This result has been strengthened to ZC3) [a15].
There are several useful and interesting extensions of the above conjectures. Suppose that is a normal subgroup of index
in
. Then
can be represented by
-matrices over
. Any torsion unit
of
that is mapped by the natural homomorphism
to
gives rise to a torsion matrix
. Here,
denotes the subgroup of the general linear group
consisting of the matrices
that are mapped by the augmentation homomorphism
, when applied to each entry, to the identity matrix. Thus, ZC1) translates to the question about diagonalization of
in
: Is a torsion matrix
, where
is a finite group, conjugate in
to a matrix of the form
,
?
This was answered positively in [a16] for -groups (cf. also
-group). See [a1] for an explicit example of a matrix
that cannot be diagonalized but for which
. Such a matrix
exists for a finite nilpotent group
and some
if and only if
has at least two non-cyclic Sylow
-subgroups [a1] (cf. also Sylow subgroup).
However, it was proved in [a6] that if and
is finite Abelian, then
is conjugate in
to
. This has been extended to
in [a8], bridging the gap between
and
.
The Zassenhaus conjectures and the isomorphism problem have also been studied for infinite groups . The statements remain the same and the group
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 can be stably diagonalized to
,
. This has been proved [a7] to be true for
-elements
when
is nilpotent. The isomorphism problem also has a positive answer for finitely-generated nilpotent groups of class
, 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 ![]() |
[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 ![]() ![]() |
[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) |
Zassenhaus conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zassenhaus_conjecture&oldid=43797