On Extension Problems Of Morphisms In Fusion Systems

The aim of this paper is to give a new criterion about Sylow intersections by using morphism extension.An exploration was made on some basic properties of the maximal Sylow intersections and strong Sylow intersection subgroups.Besides,the relationship between them is established.Furthermore,a characterization was pre-sented on essential subgroups and domestic subgroups which were defined by Craven in 2010.In particular,a useful criterion is presented for essential subgroups and do-mestic subgroups.Finally,a brief proof for Alperin's fusion theorem is obtained by using morphism extension.The main conclusions of this paper are as follows:Theorem A.Let G be a finite group,where p is a prime and P?Sylp(G).If Q is a proper subgroup of P and(?):Q?P in FP(G),then the following properties hold:(1)(?)can always extend to a Sylow intersection.In particular,if(?)can not extend,then Q must be a Sylow intersection of P.Conversely,if Q is a maximal Sylow intersection in P,then there is a morphism?:Q?P in Fp(G)that can not extend.(2)Let?= cg.Then?can not extend if and only if for every element x?CG(Q)we have Q = P?P(xg)-1.Equivalently,a subgroup D?P is a strong Sylow intersection if and only if there is a morphism(?):D?P in FP(G)that can not extend.(3)If Q is a maximal Sylow intersection in P,then Q is a strong Sylow inter-section.Theorem B.Let F be a saturated fusion system on a finite p-group P,where p is a prime and suppose that Q is fully F-normalized.(1)Q?Fe if and only if there is a F-isomorphism of Q that can not strongly extend.(2)Q?Fd if and only if there is a F-isomorphism of Q that can not extend.