Subgroups of division rings (Q2288240)

The paper under review is about the multiplicative subgroups of division algebras. The main focus is quaternion algebras over real number fields. The symbol \((a,b)_F\) denotes the algebra generated over a field \(F\) by \(i\) and \(j\) subject to the relations \(i^2=a\), \(j^2=b\) and \(ij=-ji\) for some \(a,b \in F^\times\). The paper contains several quite technical results, one of which is Theorem 11 which states that when \(F\) is a real number field, \((-1,-1)_F\) has \(\operatorname{SL}(2,5)\) as a multiplicative subgroup if and only if \((-1,-1)_F\) contains \((-1,-1)_{\mathbb{Q}(\sqrt{5})}\) as a subring, and \((-1,-1)_F\) has the binary octahedral group \(B\) of order 48, also called the Clifford group (this is the group of order 48 that is isoclinic to but not isomorphic to \(\operatorname{GL}(2,3)\)), if and only if \((-1,-1)_F\) contains \((-1,-1)_{\mathbb{Q}(\sqrt{2})}\) as a subring. The authors used of Magma in their proofs, and they provide the codes.