  We introduce an equivariant algebraic cobordism theory \Omega^G for algebraic
varieties with G-action, where G is a split diagonalizable group scheme over a
field k. It is done by combining the construction of the algebraic cobordism
theory \Omega by F. Morel and M. Levine, with the notion of (G, F)-formal group
law with respect to a complete G-universe and complete G-flag F as introduced
by M. Cole, J. P. C. Greenlees and I. Kriz. In particular, we use their
corresponding representing ring L_G(F) in place of the Lazard ring L. We show
that localization property and homotopy invariance property hold in \Omega^G.
We also prove the surjectivity of the canonical map from L_G(F) to
\Omega^G(Spec k). Moreover, we give some comparison results with \Omega, the
equivariant algebraic cobordism theory introduced by J. Heller and J.
Malagon-Lopez, the equivariant K-theory and Tom Dieck equivariant cobordism
theory (when k = C). In particular, we proved the equivariant Conner-Floyd
isomorphism when char k = 0. Finally, we show that our definition of \Omega^G
is independent of the choice of F.
