Why that element belongs to FC subgroup?

Question

Let $\Delta$ denote Finite Conjugate subgroup of group G. K - a field

Then

The photo above is from Passman book "Infinite group rings".

I dont understand why elements $u_{i}$ belong to $\Delta$.

Could You tell me that?

I understand everything else on that photo.

Answer 1

Since $\theta(\alpha)$ and $\theta(\gamma)$ belong to the image of $\theta$ which is contained in (equal to, actually) $K[\Delta]$, their supports must be in $\Delta$.

More explicitly, write $\theta(\alpha) = \sum_{x\in\Delta} k_x x$ and $\theta(\gamma) = \sum_{x\in\Delta} \ell_x x$. Then $\{u_1, u_2, \ldots, u_r\}$ is just the set of $x\in\Delta$ such that $k_x\neq 0$ or $\ell_x \neq 0$ in those two sums.