A note on solvable maximal subgroups in subnormal subgroups of ${\mathrm GL}_n(D)$

Abstract

Let D be a non-commutative division ring, and G be a subnormal subgroup of GLn(D). Assume additionally that the center of D contains at least five elements if n > 1. In this note, we show that if G contains a nonabelian solvable maximal subgroup, then n = 1 and D is a cyclic algebra of prime degree over the center.We note that this conjecture is not true if n = 1. Indeed, it was proved in [1] that the subgroup C * ∪ C * j is a solvable maximal subgroup of the multiplicative group H * of the division ring of real quaternions H. In this note, we show that Conjecture 1 is true for non-abelian solvable maximal subgroups of G, that is, we prove that G contains no non-abelian solvable maximal subgroups. This fact generalizes the main result in [2] and it is a consequence of Theorem 3.7 in the text.Throughout this note, we denote by D a division ring with center F and by D * the multiplicative group of D. For a positive integer n, the symbol M n (D) stands for the matrix ring of degree n over D. We identify F with F I n via the ring isomorphism a → aI n , where I n is the identity matrix of degree n. If S is a subset of M n (D), then F [S] denotes the subring of M n (D) generated by the set S ∪ F . Also, if n = 1, i.e., if S ⊆ D, then F (S) is the division subring of D generated by S ∪ F . Recall that a division ring D is locally finite if for every finite subset S of D, the division subring F (S) is a finite dimensional vector space over F . If H and K are two subgroups in a group G, then N K (H) denotes the set of all elements k ∈ K such that k −1 Hk ≤ H, i.e., N K (H) = K ∩ N G (H). If A is a ring or a group, then Z(A) denotes the center of A.

On locally solvable subgroups in division rings

“…Setting n = [D : F ], we know that D ⊗ F D op ∼ = M n (F ). Thus, by viewing M as a subgroup of GL n (F ), we conclude that M is a solvable group and the results follow from [8,Theorem 3.2] or [1].…”

“…According Lemma 2.3, we deduce that M (i) is abelian. Hence M is solvable, and we are done by [8,Theorem 3.2] or [1].…”

“…The first case implies that M is solvable, and hence [D : F ] < ∞ by [8,Theorem 3.2] or [1]. Now, suppose the second case occurs, from which it follows by [11,Theorem 1.1] that M is abelian-by-locally finite.…”