tailieunhanh - Đề tài " Periodic simple groups of finitary linear transformations "

In Memory of Dick and Brian Abstract A group is locally finite if every finite subset generates a finite subgroup. A group of linear transformations is finitary if each element minus the identity is an endomorphism of finite rank. The classification and structure theory for locally finite simple groups splits naturally into two cases—those groups that can be faithfully represented as groups of finitary linear transformations and those groups that are not finitary linear. This paper completes the finitary case. We classify up to isomorphism those infinite, locally finite, simple groups that are finitary linear but not linear. . | Annals of Mathematics Periodic simple groups of finitary linear transformations By J. I. Hall Annals of Mathematics 163 2006 445 498 Periodic simple groups of finitary linear transformations By J. I. Hall In Memory of Dick and Brian Abstract A group is locally finite if every finite subset generates a finite subgroup. A group of linear transformations is finitary if each element minus the identity is an endomorphism of finite rank. The classification and structure theory for locally finite simple groups splits naturally into two cases those groups that can be faithfully represented as groups of finitary linear transformations and those groups that are not finitary linear. This paper completes the finitary case. We classify up to isomorphism those infinite locally finite simple groups that are finitary linear but not linear. 1. Introduction A group G is locally finite if every finite subset S is contained in a finite subgroup of G. That is every finite S generates a finite subgroup S . This paper presents one step in the classification of those locally finite groups that are simple. We shall be particularly interested in locally finite simple groups that have faithful representations as finitary linear groups the finitary locally finite simple groups. Let V be a left vector space over the field K. For us fields will always be commutative. Thus Endy V acts on the right with group of units GLy V . The element g G GLy V is finitary if V g 1 V g has finite K-dimension. This dimension is the degree of g on V degy g limy V g . Equivalently g is finitary on V if and only if limy V Cy g is finite where Cy g ker g 1 . In this case dimy V Cy g degy g. The invertible finitary linear transformations of V form a normal subgroup of GLy V that is denoted FGLy V the finitary general linear group. A Partial support provided by the NSA. 446 J. I. HALL group G is finitary linear sometimes shortened to finitary if it has a faithful representation p G FGLk V for some vector space V over