Let $S$ be a scheme and let $X := \mathrm{Spec}(\mathscr{O}\_S[t, t^{-1}])$ be the underlying $S$-scheme of the $S$-group scheme $(\mathbb{G}\_m)\_S$. Is there only one structure of a commutative $S$-group scheme on $X$ for which $t = 1$ is the identity section?



Is there a unique commutative group structure on $\mathbb{G}_m$?
Yes if $S$ is reduced, and no otherwise. The case of a field is classical (ultimately because $k[t,1/t]^{\times} = k^{\times}\cdot t^{\mathbf{Z}}$ for fields $k$), and I assume you are familiar with that. So in general if $S$ is reduced then by passing to the case of affine $S$ and then noetherian $S$ (by the usual limit business) we have $S = {\rm{Spec}}(R)$ with just finitely many generic points. At each generic fiber the desired equality of hypothetical group laws is known by the settled case of fields, so it holds globally by reducedness of the base.


In the non-reduced case the presence of "extra" units creates non-homomorphic automorphisms of the pointed scheme. Say $S = {\rm{Spec}}(R)$ and $r \in R$ is a nonzero nilpotent element satisfying $r^2=0$. Since $r+t \in R[t,1/t]^{\times}$ (with inverse $(1-r/t)/t = 1/t - r/t^2$), we get an automorphism of the pointed scheme $({\rm{GL}}\_1,1)$ via $t \mapsto (1-r)(r+t)=r+(1-r)t$. Transferring the usual group law through this automorphism provides another group law.


But maybe you are not asking the question you intend. There are very strong results in SGA3 concerning the infinitesimal deformation theory of relative tori and rigidity results thereof. What is your motivation for the question posed? 


