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 \in \text{supp}(u) $
- $ \text{supp}(u) \subseteq \Delta(G) $
- $ \text{conj}(u) = \text{id}_{G/\Delta^+(G)} $
- $ \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) $