Another proof of B. Sury's theorem.
The structure of normal subgroups of the abstract group \(G(k)\) for an absolutely simple, simply connected algebraic group \(G\) over a global field \(k\), is essentially known excepting the \(k\)-anisotropic groups of outer type \(A_n\). This group can be described as \[ \text{SU}(D,\sigma):=\{d\in D^*:dd^\sigma=1,\;N_{\text{red}}(d)=1\} \] for a division algebra \(D\) with an involution \(\sigma\) of the second kind (that is, \(\sigma\) restricts to the nontrivial automorphism of the center \(K\) of \(D\) -- so, \(K\) is a quadratic extension of \(k\)). As \([\text{U}(D,\sigma),\text{U}(D,\sigma)]\) is a normal subgroup of \(\text{SU}(D,\sigma)\), where \(\text{U}(D,\sigma)=\{d\in D:dd^\sigma=1\}\), the first natural question is whether \(\text{SU}(D,\sigma)=[\text{U}(D,\sigma),\text{U}(D,\sigma)]\) in analogy with the classical result \(\text{SL}(1,D)=[D^*,D^*]\) over global fields. In fact, one might consider more general fields which admit such division algebras with involutions of the second kind and, ask for the determination of the quotient group \(\text{SU}(D,\sigma)/[\text{U}(D,\sigma),\text{U}(D,\sigma)]\). In [Proc. Am. Math. Soc. 134, No. 2, 351-354 (2006; Zbl 1088.16016)], the reviewer and \textit{B. A. Sethuraman} showed that this quotient can be infinite for some function fields over number fields. However, the question remains open for general degree division algebras over global fields. In [Arch. Math. 90, No. 6, 493-500 (2008; Zbl 1225.16007)], the reviewer had computed the quotient when \(D\) is a quaternion division algebra and showed it to be nontrivial in general. In the paper under review, the authors give another proof which could perhaps prove useful in treating more general degrees later. The second author has been an authority in the subject for several decades and has contributed copiously. Thus, it is always desirable to see a contribution by him on this even if it is another proof of a known theorem.
scientific article; zbMATH DE number 5660152