Arch. Math., Vol. 67, 183-191 (1996) 0003-889X/96/6703-0183 $ 3.30/0 9 1996 Birkh/iuser Verlag, Basel Clifford theory on the Burnside ring By I. LIZASOAIN and G. OCHOA*) 1. Introduction. All groups considered in this paper are finite. Let G be a group and C(G) the set of the conjugacy classes of subgroups of G. Denote by (U)G the conjugacy class of a subgroup U of G. Let Q (G) be the Burnside ring of G (that is, the Grothendieck ring of finite G-sets). The set of (isomorphism classes of) G-sets {G/U; (U)~ ~ C (G)} is a ;g-basis of f2 (G) (see tom Dieck [2] and Dress [4]). If H < G, the restriction Resg: f2 (G) ~ f~ (H), which maps the G-set X to the H-set Xn, is a homomorphism of rings and the induction Indg' f2(H) --. f2(G) is a homo- morphism of Y-modules. These maps are in correspondence with the analogous in the ring of complex characters, R(G), through the canonical ring homomorphism r: ~2(G) ~ R (G) mapping a G-set X to the character of the permutation module having X as a 2Lbasis. Also, Mackey's formula and Frobenius' reciprocity are valid in ~2(G) (see Dress [5] and Ochoa [10]). Nevertheless, no version of Clifford theory for Burnside rings appears in the bibliogra- phy. Maybe the reason for this gap is that the scalar product obtained from the one of characters, which gives (G/U, G/V) = card {UxV; x ~ G}, is regular if and only if G is cyclic (see Th6venaz [14]). These considerations lead us to the introduction of a new scalar product and a new induction (called elevation) which allows us to achieve a Clifford theoryJor Burnside rings, whose behaviour corresponds in many aspects to the Clifford theory for characters. Furthermore, in the third section, we consider some natural concepts in ~2(G) derived from the elevation (primitive, monomial) and we determine the kinds of glZoups whose permutation representations have some special properties. In particular, we obtain a characterization of nilpotent groups. 2. Clifford theory. Consider the symmetric bilinear form (,): ~ (G) x Q (G) ~ 2g given by (G/U,G/V)={IO if(U)a=(V)G if (U)~ + (V)~" With respect to this scalar product, the classical induction is not the adjunction of the restriction. So the following definition makes sense. *) The authors are supported by Projecto "Acciones de Grupos finitos" of Gobierno de Navarra and DGICYT PB 90-0414-C03-023