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Let $G$ be a group, $R$ a commutative ring with $1$ and $ u \in \text{N}_{\text{U}(RG)}(G) $ with the following properties:

  1. $ 1 \in \text{supp}(u) $
  2. $ \text{supp}(u) \subseteq \Delta(G) $
  3. $ \text{conj}(u) = \text{id}_{G/\Delta^+(G)} $
  4. $ \text{conj}(u) \in \text{Aut}(G) $ has finite order

How can I show that $ G \cap \langle u \rangle = \{1\} $ ?

$ \Delta(G) = $ FC-Center of G
$ \Delta^+(G) = $ Tosion elements of $ \Delta(G) $

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What's $\text{supp}(u)$? –  Alexander Gruber Dec 9 '12 at 18:34
    
When $ u = \sum_{g \in G} r_g g $ then $ \text{supp}(u) = \{ g \in G | r_g \neq 0 \} $. –  Boris Dec 10 '12 at 11:09
    
Then, in (1), it'd perhaps be a good idea to write $\,1_G\in supp(u)\,$ , to distinguish the ring and group units. –  DonAntonio Dec 14 '12 at 16:08
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