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Group algebras whose group of units is powerful

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 Added by Victor Bovdi
 Publication date 2009
  fields
and research's language is English
 Authors V.A. Bovdi




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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.



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58 - V.A. Bovdi , M.A. Salim 2016
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.
296 - V.A. Bovdi , L.G. Kovacs 2007
Let p be a prime, K a field of characteristic p, G a locally finite p-group, KG the group algebra, and V the group of the units of KG with augmentation 1. The anti-automorphism gmapsto g^{-1} of G extends linearly to KG; this extension leaves V setwise invariant, and its restriction to V followed by vmapsto v^{-1} lives an automorphism of V. The elements of V fixed by this automorphism are called unitary; they form a subgroup. Our first theorem describes the K and G for which this subgroup is normal in V. For each element g in G, let bar{g} denote the sum (in KG) of the distinct powers of g. The elements 1+(g-1)hbar{g} with g,hin G are the bicyclic units of KG. Our second theorem describes the K and G for which all bicyclic units are unitary.
190 - V.A. Bovdi , A.B. Konovalov 2007
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