The object of this article is the Burnside ring Ω(G) for a finite group G. If G is soluble we show how the group of ring automorphisms Aut(Ω(G)) may be calculated purely from the knowledge of the subgroup lattice of G. The results on Aut(Ω(G)) for an abelian group G by Krämer [6] and results of Lezaun [7] for dihedral and some special metacyclic groups follow as special cases. For general facts on Burnside rings we refer to [2, §80].

The paper is organized as follows: In Section 1 we recall some more or less well known facts about the table of marks, the ghost ring of Ω(G) and Aut(Ω(G)). We introduce the group of normalized automorphism Autn(Ω(G)) as the automorphisms which stabilize the regular G – set G/l. In (2.4) we reduce the study of Aut(Ω(G)) for soluble groups to the study of Autn(Ω(G)), extending a result of Krämer [6] from nilpotent groups to soluble groups. Any automorphism a of Ω(G) induces in a natural way – via the ghost ring – a bijection σ* on the set V(G) of conjugacy classes of subgroups of G. In (3.1) it is shown that this is indeed an automorphism of V(G) considered in a special way as a partially ordered set, provided σ is augmented and G is soluble. In (3.2) we characterize those σ ∈ Aut(Ω(G)), which send transitive G-sets to transitive G-sets, in terms of the map σ*. In (3.6) we give an explicit constructive description of Autn(Ω(G)) in terms of the automorphisms of V(G) as a partially ordered set; i.e. for a fixed group it allows one to compute explicitly Autn(Ω(G)) with the help of a computer algebra system.