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Register a new userRational group algebras of finite groups: from idempotents to units of integral group rings
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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 ring $Z G$ of a finite nilpotent group $G$ has 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 elements $e(G,H,K)$ 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 such $e(G,H,K)$, with $(H,K)$ strong Shoda pairs in subgroups of $G$.
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Using the Luthar--Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups $Co_1$, $Co_2$ and $Co_3$.
During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $U (Z G)$ of the integral group ring $Z G$ of a finite group $G$. These constructions rely on explicit constructions of units in $Z G$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $Q G$. The latter relies on explicit constructions of primitive central idempotents and the rational representations of $G$. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.
The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $mathbb Z G$ , i.e. the prime graph of the normalised unit group of $mathbb Z G$ coincides with that one of the group $G$. In this note we prove for finite groups $G$ a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups $G$ whose order is divisible by at most three primes and show that the Gruenberg - Kegel graph of such groups coincides with the prime graph of $G$.
A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.
We give a full description of locally finite p-groups G such that the normalized group of units V(FG) of the group algebra FG over a field F of characteristic p has exponent 4.
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