Presentation of the group of units of ZD-16

Abstract

One of the Open Problems listed by Sehgal in [9] asks for obtaining presentations by generators and relations of the group of units ZG∗ of the integral group ring ZG for some finite groups G. This problem has been solved up to finite index for seven of the nine non abelian groups of order 16. (Recall that Higman described the structure of ZG∗ for G abelian [4].) Namely it is well known since Higman that Z(Q8 ×C2)∗ = Q8 ×C2 [9] (see notation below); if G is D8 × C2, C4 C4, Q16 or H then ZG∗ contains a subgroup of finite index that is a direct product of free groups (see [5] and [6]). Furthermore, a concrete direct product of free groups F of ZG∗ with minimal index in ZG∗ has been computed in [8]; a presentation of ZG∗ has been obtained in [7] for the groups D and D+16. Thus at this moment there are two groups of order 16 for which no presentation of ZG∗ by generators and relations is known, namely these groups are D16 and D−16. Here is the notation: Cn = Cyclic group of order n. Q4n = Quaternion group of order 4n. D2n = Dihedral group of order 2n.