n this paper we generate a G-algebra from a non-empty set and we obtain the quotient G-algebra via normal subalgebra. Furthermore, we prove a fundamental theorem on homomorphism for G-algebra. We prove that every G-algebra satisfying the associative law is a 2-group. We also show that every BP-algebra is a G-algebra and introduce a necessary condition for which the converse will be true. Finally, we introduce the notion of a left-right (resp. right-left) derivation of G-algebra. We show that the composition of two derivations is a derivation and we investigate some related properties.