A group theoretically based method of quantization is studied. The linear space F of classical objects considered as an Abelian group is centrally extended by the Abelian group of complex numbers. The different possible factors of this extension determine the different commutation relations of the corresponding quantum theory. Therefore, properties of factors and reasonable restrictions on them (analyticity or invariance under a group of motions) are studied. The method is illustrated for the case of F being the space of scalar functions of an index set I with different choices of I: \(I=\{1,2,...,n\}\) leads to time independent quantum mechanics; \(I={\mathbb{R}}\times \{1,2,...,n\}\) to time dependent quantum mechanics; \(I=Minkowski\) space leads to quantum field theory which can be extended by additional internal degrees of freedom.