Rational Group Algebras of Finite Groups: From Idempotents to Units of Integral Group Rings
Rational Group Algebras of Finite Groups: From Idempotents to Units of Integral Group Rings
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DOI：
10.1007/s10468-010-9244-4
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发表时间：
2010-01
影响因子：
0.6
通讯作者：
E. Jespers;G. Olteanu;Á. del Río
中科院分区：
文献类型：
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作者：
E. Jespers;G. Olteanu;Á. del Río
给出了有限幂零群的有理群代数的正交本原幂等元的完备集的一个明确的无特征标的构造，并给出了这类代数的Wedderburn分解的完整描述.一个直接的后果是一个众所周知的结果罗盖特的舒尔指数的简单组成部分的群代数的有限幂零群。作为应用，我们得到了有限幂零群G的整群环的单位群有一个由三个幂零群生成的有限指数子群，并给出了它们的生成元的明确描述.另一个应用是单位群中自由子群的新构造。在所有的构造中，子群对（H，K）（称为强Shoda对）和显式构造的中心元（G，H，K）起着至关重要的作用。对任意有限群，证明了有理群代数的本原中心幂等元是这样的（G，H，K）的有理线性组合，且（H，K）强Shoda对在G的子群中.
We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ringof a finite nilpotent groupGhas a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators. Another application is a new construction of free subgroups in the unit group. In all the constructions dealt with, pairs of subgroups (H,K), called strong Shoda pairs, and explicit constructed central elementse(G,H,K) play a crucial role. For arbitrary finite groups we prove that the primitive central idempotents of the rational group algebras are rational linear combinations of suche(G,H,K), with (H,K) strong Shoda pairs in subgroups ofG.