  We give some statements that are equivalent to the existence of group
inverses of Peirce corner matrices of a $2 \times 2$ block matrix and its
generalized Schur complements. As applications, several new results for the
Drazin inverses of the generalized Schur complements and the $2 \times 2$ block
matrix are obtained and some of them generalize several results in the
literature.
