# Inner automorphism of a group-ring

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 $\Delta(G)$

-
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