AbstractsPhysics

Let G be a group and P a Sylow p -subgroup of G . A fusion system of G on P , denoted by FP (G) , is the category with objects; subgroups of P , and morphisms induced by conjugation in G . This thesis gives a brief introduction to the theory fusion systems. Two classical theorems of Burnside and Frobenius are stated and proved. These theorems may be seen as a starting point of the theory of fusion systems, even though the axiomatic foundation is due to Puig in the early 1990's. An abstract fusion system F on a p -group P is dened and the notion of a saturated fusion system is discussed. It turns out that the fusion system of any nite group is saturated, but the converse; that a saturated fusion system is realizable on a nite group, is not always true. Two versions of Alperin's fusion theorem are stated and proved. The first one is the classical formulation of Alperin and the second one, due to Puig, a version stated in the language of fusion systems. The differences between these two are investigated. The fusion system F of <em>GL₂</em>(3) on the Sylow 2-subgroup isomorphicto <em>SD₁₆</em> is determined and the subgroups generating F are found.