A Criterion for the Triviality ofG(D) and Its Applications to the Multiplicative Structure ofD

Multiplicative groups of division rings

“…Finally, we give some additional results concerning the group G(D) and maximal subgroups of D * . For the proofs of these theorems see [69]. (1) D contains an absolutely irreducible nilpotent subgroup;…”

“…In [46] Keshavarzipour and Mahdavi-Hezavehi proved that if the deg(D) = p m for some prime p and µ p ⊂ Z(D) then triviality of G(D) forces that D is the quaternion division algebra. However, a general criterion for the triviality of G(D) was provided in [69]. This section is devoted to presenting this criterion and then applying this criterion to achieve the main result of [25].…”

“…By combining the above observation, Theorem 8.9, Corollary 8.7 and Lemma 3 of [71] which asserts that every divisible field of characteristic zero contains all primitive roots of unity, we obtain: Finally, we give some additional results concerning the group G(D) and maximal subgroups of D * . For the proofs of these theorems see [69].…”

“…[69]). Let D be an F -central division algebra of prime degree p. Then D is a symbol algebra if and only if D * contains a non-abelian nilpotent subgroup.…”