Existence of Nonabelian Free Subgroups in the Maximal Subgroups of GLn(D)

• 作者: H.R. Dorbidi R. Fallah-Moghaddam M. Mahdavi-Hezavehi
• 来源期刊: 代数集刊:英文版
• 学科: 数学
• 关键词: 极大子群 Gln 非交换 自由 阿贝尔 有限维 非循环 伽罗瓦
• 年，卷（期）: 2014,（3）
• 页码范围: 483-496
• 页数: 14页
• 分类号: O152.1
• 语种: 中文

摘要

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of CLn (D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NCLn(D)(K*) = M, K* △ M, K/F is Galois with Gal(K/F) ≌ M/K*, and F[M] = in(D). In particular, when F is global or local, it is proved that if ([D : F], Char(F)) = 1, then every non- abelian maximal subgroup of GL1 (D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5.