  A Smarandache multi-space is a union of $n$ spaces $A_1,A_2,..., A_n$ with
some additional conditions holding. Combining classical of a group with
Smarandache multi-spaces, the conception of a multi-group space is introduced
in this paper, which is a generalization of the classical algebraic structures,
such as the group, filed, body, $...$, etc.. Similar to groups, some
characteristics of a multi-group space are obtained in this paper.
