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 Subsequently, they also gave a counterexample to ZC2) (unpublished), which appears in a modified form in [a5].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) was disproved in 2018 [a19]. 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 -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 -adic -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