Extensions Of Source Algebras

The definitions of hyperfocal subgroup and basic Morita equivalence are intro?duced by L.Puig.Let p be a prime,G,G’finite groups,and R a normal p-subgroup of G such that |G:CG(R)| is a p-power,and R’ a normal p-subgroup of G’ such that |G’:CG,(R’)| is a p-power.Let b and b’ be blocks(idempotents)of G and G’with maximal local pointed groups P-and P’γ’ respectively.Assume that Qs and Q’δ’ are local pointed groups of QG and OG’ respectively.Denote the set of G’-exomorphisms from Qs to Qs by EG(Qs).For an essential pointed group Qs,denote the unique minimal element of the set of normal subgroups of EG(Qs)which are not p’-group by XG{Qδ).Similar notation for EG’(Q’δ’)and XG,(Q’δ’).In this paper,firstly,we prove that a hyperfocal subgroup of the block b is the image of the one of the block b under the canonical surjective G→G G= G/R,where b is the block of G determined by b.In particular,the block b is nilpotent if and only if the block b is nilpotent.Secondly,assume that there is a basic Morita equivalence between blocks b and b’ determining a group isomorphism σ:P≌ P’ which can be lifted in a group isomorphism σ:P≌P’ inducing group isomorphisms EG(Pγ)≌ EG’(P’γ’)and XG(Qδ)≌XG,(Q’δ’)for some representatives Qs and Q’δ’,respectively contained in Pγ and P’γ’,and fulfilling σ(Q)= Q’,of any G-and G’-conjugacy classes of es-sential pointed groups on OGb and on OG’b’,we show that there is a basic Morita equivalence between blocks b and b’ determining σ.