Centrally large subgroups of finite p-groups

Let S be a finite p-group. We say that an abelian subgroup A of S is a large abelian subgroup of S if |A|⩾|A∗| for every abelian subgroup A∗ of S. We say that a subgroup Q of S is a centrally large subgroup, or CL-subgroup, of S if |Q|⋅|Z(Q)|⩾|Q∗|⋅|Z(Q∗)| for every subgroup Q∗ of S. The study of large abelian subgroups and variations on them began in 1964 with Thompson's second normal p-complement theorem [J.G. Thompson, Normal p-complements for finite groups, J. Algebra 1 (1964) 43–46]. Centrally large subgroups possess some similar properties. In 1989, A. Chermak and A. Delgado [A. Chermak, A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989) 907–914] studied several families of subgroups that include centrally large subgroups as a special case. In this paper, we extend their work to prove some further properties of centrally large subgroups. The proof uses an analogue for finite p-groups of an application of Borel's Fixed Point Theorem for algebraic groups.