Integral group rings: Groups of units and classical K-theory

Abstract

This survey contains results obtained in this area from the second half of the sixties to the present time. In the group of units of the group ring, normal periodic subgroups, elements of finite order, free subgroups, congruence subgroups, questions of conjugacy of finite subgroups, and matrix representations are considered. Moreover, questions connected with the calculation of the groups K0, K I for group rings are discussed with a description of the structure of projective modules, groups of invertib!e matrices, etc.

Integral Group Ring of the Mathieu Simple GroupM23

“…The Luthar-Passi method proved to be useful for groups containing non-trivial normal subgroups as well. Also some related properties and some weakened variations of the Zassenhaus conjecture as well can be found in [1,22] and [3,20]. For some recent results we refer to [5,7,15,16,17,18].…”

Torsion units in integral group rings of Janko simple groups

“…Denote by ν nt = ν nt (u) = g∈C nt α g , the partial augmentation of u with respect to C nt . From the Berman-Higman Theorem (see, for example, [1]) one knows that tr(u) = ν 1 = 0, and clearly (1) C nt ∈C ν nt = 1.…”

Integral group ring of the Mathieu simple groupM 12

“…For some recent results we refer to [5,7,12,14,13,15]. Also, some related properties and some weakened variations of the Zassenhaus conjecture can be found in [1,18] and [3,16].…”

“…For some recent results we refer to [5,7,12,14,13,15]. Also, some related properties and some weakened variations of the Zassenhaus conjecture can be found in [1,18] and [3,16].First of all, we need to introduce some notation. By #(G) we denote the set of all primes dividing the order of G. The Gruenberg-Kegel graph (or the prime graph) of G is the graph π(G) with vertices labeled by the primes in #(G) and with an edge from p to q if there is an element of order pq in the group G. In [16] W. Kimmerle proposed the following weakened variation of the Zassenhaus conjecture:…”