We characterize the restricted Lie algebras L whose restricted universal enveloping algebra u(L) is Lie metabelian. Moreover, we show that the last condition is equivalent to u(L) being strongly Lie metabelian

Abstract. We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M 12 . As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs. Introduction, conjectures and main resultsLet V (ZG) be the normalized unit group of the integral group ring ZG of a finite group G. A long-standing conjecture of H. Zassenhaus (ZC) says that every torsion unit u ∈ V (ZG) is conjugate within the rational group algebra QG to an element in G.For finite simple groups the main tool for the investigation of the Zassenhaus conjecture is the Luthar-Passi method, introduced in [17] to solve it for A 5 . Later M. Hertweck in [14] extended the Luthar-Passi method and applied it for the investigation of the Zassenhaus conjecture for P SL(2, p n ). The Luthar-Passi method proved to be useful for groups containing non-trivial normal subgroups as well. 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:(KC) If G is a finite group then π(G) = π(V (ZG)). In particular, in the same paper W. Kimmerle verified that (KC) holds for finite Frobenius and solvable groups. Note that with respect to the so-called p-version of the Zassenhaus conjecture the investigation of Frobenius groups was completed by M. Hertweck and the first author in [4]. In [6,7,8] (KC) was confirmed for sporadic simple groups M 11 , M 23 and some Janko simple groups.Here we continue these investigations for the Mathieu simple group M 12 . Although using the Luthar-Passi method we cannot prove the rational conjugacy for torsion units of V (ZM 12 ), our main result gives a lot of information on partial 1991 Mathematics Subject Classification. Primary 16S34, 20C05, secondary 20D08.

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