Sylow intersections in finite groups
The meet semilattice generated by \(\text{Syl}_p(G)\) within the subgroup lattice of a finite group \(G\) is studied, where \(p\) is a fixed prime. The main result implies that if \(p\) is an odd non-Mersenne prime, \(G\) is a finite \(p\)-solvable group, and \(P_1\), \(P_2\), \(P_3\) are Sylow \(p\)-subgroups of \(G\), then \(G\) possesses a Sylow \(p\)-subgroup \(P_4\) such that \(P_1\cap P_4=P_1\cap P_2\cap P_3\). It is shown that if \(p=2\) or \(p\) is Mersenne, \(G\) is a finite \(p\)-solvable group, then the intersection of any four Sylow \(p\)-subgroups can be obtained as the intersection of three Sylow \(p\)-subgroups with one Sylow \(p\)-subgroup in common.
scientific article; zbMATH DE number 1214152
scientific article