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Let $\Xi$ stand for a finite abelian spin structure group of four-dimensionalsuperstring theory in free fermionic formulation whose elements are64-dimensional vectors (spin structure vectors) with rational entries belongingto $\rbrack -1,\, 1\rbrack $ and the group operation is the $mod\, \, 2 $ entryby entry summation $\oplus $ of these vectors. Let $B=\{b_i,\, i= 1,\cdots,k+1\}$ be a set of spin structure vectors such that $b_i$ have only entries 0and 1 for any $\, i= 1,\cdots ,k$, while $b_{k+1}$ is allowed to have anyrational entries belonging to $\rbrack -1,\, 1\rbrack $ with even $N_{k+1}$,where $N_{k+1}$ stands for the least positive integer such that$N_{k+1}b_{k+1}= 0\,mod\,2$. Let $B$ be a basis of $\Xi$, i.e., let $B$generate $\Xi$, and let $\Lambda_{m, n}$ stand for the transformation of $B$which replaces $b_n$ by $b_m\oplus b_n$ for any $m \ne k+1$, $n \ne 1$, $m \nen$. We prove that if $B$ satisfies the axioms for a basis of spin structuregroup $\Xi$, then $B'=\Lambda_{m, n}B$ also satisfies the axioms. Since thetransformations $\Lambda_{m,n}$ for different $m$ and $n$ generate allnondegenerate transformations of the basis $B$ that preserve the vector $b_1$and a single vector $ b_{k+1} $ with general rational entries, we conclude thatthe axioms are conditions for the whole group $\Xi$ and not just conditions fora particular choice of its basis. Hence, these transformations generate thediscrete symmetry group of four-dimensional superstring models in freefermionic formulation.Comment: 10 pages, MIU-THP-94/6

On transformations preserving the basis conditions of a spin structure group in four-dimensional superstring theory in free fermionic formulation