On the non-triviality of G(D) and the existence of maximal subgroups of GL1(D)

Let D be an F-central division algebra of index n. Here we investigate a conjecture posed in [R. Hazrat et al., Reduced K-theory and the group G(D)=D*/F*D′, in: Algebraic K-theory and its Applications, Trieste, 1997, pp. 403–409] that if D is not a quaternion algebra, then the group G0(D)=D*/F*D′ is non-trivial. Assume that either D is cyclic or F contains a primitive pth root of unity for some prime pn. Using Merkurjev–Suslin Theorem, it is essentially shown that if none of the primary components of D is a quaternion algebra, then G(D)=D*/RND/F(D*)D′≠1. In this direction, we also study a conjecture posed in [S. Akbari, M. Mahdavi-Hezavehi, J. Pure Appl. Algebra 171 (2002) 123–131] or also [M. Mahdavi-Hezavehi, J. Algebra 271 (2004) 518–528] on the existence of maximal subgroups of D*. It is shown that if D is not a quaternion algebra with i(D)=pe, then D* has a maximal subgroup if either of the following conditions holds: (i) F has characteristic zero, or (ii) F has characteristic p, or (iii) F contains a primitive pth root of unity.