Group inverse for a class 2 × 2 block matrices over skew fields

Suppose K is a skew field. Let K n × n denote the set of all matrices over K. For A ∈ K n × n , the matrix X ∈ K n × n is said to be the group inverse of A, if it holds that AXA = A , XAX = X , AX = XA . In this paper, we give the existence and the representation of the group inverse for block matrix M = A A B 0 ( A , B ∈ K n × n , A 2 = A ) over skew fields, and give the existences and the representations of the group inverse for other block matrices.