Block-diagonal characterization of locally finite simple groups of p-type

In this paper, we prove the following characterization of LFS-groups of p-type: Let G be a locally finite group and S be a local system for G such that S subset of P and for all S is an element of S: (i) {T is an element of S(S)vertical bar V-T(j) is block-diagonal for S for j = 1, ..., n(T)} is a local system for G. (ii) there exists F-S is an element of S(S) such that (a) V-FS(j) is block-diagonal for S for j = 1, ..., n(FS), (b) [V-FS(j), S-i*] not equal 0 for i = 1, ..., n(S) and j = 1, ..., n(FS), (c) for j = 1, ..., n(FS), Sigma(nS)(i=1) n(SFS)(ij) dim(KFSj) K-S(i), K-FS(j) = 2 where n(SFS)(ij) is the number of composition factors C for (S) over cap on V-FS(j) with [C, S-i*] not equal 0. Then G/LSol(G) is an LFS-group of p-type and [G, LSol(G)] = O p(G). We use this theorem to construct a general family of LFS-groups of p-type.