On groups with extremal \(p\)-blocks
Remember that a finite group \(G\) is called \(p\)-closed if it has a normal Sylow \(p\)-subgroup and it is called \(p\)TI if two distinct Sylow \(p\)- subgroups intersect trivially. It is well known that a \(p\)-closed group has only \(p\)-blocks of maximal defect and a \(p\)TI-group has only blocks of maximal or zero defect. Easy examples show that in neither case the converse holds true. The purpose of this note is to give additional group theoretical properties which together with the block properties characterize the \(p\)-closure resp. \(p\)TI property. More precisely, the authors prove: \(G\) is \(p\)-closed if and only if \(G\) has only \(p\)-blocks of maximal defect and secondly, the intersection of two different Sylow \(p\)-subgroups of \(G\) is always cyclic. By a result of \textit{J.-P. Zhang} [J. Algebra 118, 129-139 (1988; Zbl 0652.20014)] the second condition may be replaced by the property: the center of the intersection of two different Sylow \(p\)-subgroups \(P\) and \(Q\) is not a Sylow subgroup of \(C_ G(P\cap Q)\). A similar result holds in the \(p\)TI case. The proofs are based on the classification of finite simple groups.
scientific article; zbMATH DE number 62233
scientific article