  In this paper, we introduce a weak group inverse (called the WG inverse in
the present paper) for square matrices of an arbitrary index, and give some of
its characterizations and properties. Furthermore, we introduce two orders: one
is a pre-order and the other is a partial order, and derive several
characterizations of the two orders. At last, one characterization of the
core-EP order is derived by using the WG inverses.
