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Two algebras 31 and S3 over the same field F are called equivalent (or simply isomorphic) if it is possible to establish between their quantities a (1-1) correspondence such that if any quantities x and y of 31 correspond to X a n d F of S3, then x+y, xy and ax correspond to X + F, XY, aX respectively for every a of F. We shall consider two generalized quaternion division algebrasf 31 and S3 over the field of all rational numbers, R. Let 31 be given by

A necessary and sufficient condition for the non-equivalence of any two rational generalized quaternion division algebras