# generator of a subring of integral group ring

Let $$G$$ be a cyclic group of order $$n$$ and $$K\leq AutG$$ be a subgroup of the automorphism group of $$G$$. We denote the orbits of the natural action of $$K$$ on $$G$$ by $$O_1,\cdots, O_s$$. Let $$\underline{X}_i=\sum_{x\in O_i}x$$ be the sum of elements in each orbit in the integral group ring $$\mathbb{Z}G$$. Then the $$\mathbb{Z}$$-span of $$\underline{X}_i$$'s is a subring $$\mathcal{A}$$ of $$\mathbb{Z}G$$. Is $$\mathcal{A}$$ always generated by some $$\underline{X}_i$$ as a ring over $$\mathbb{Z}$$?

Not when $$n=4$$ and $$G$$ is the full automorphism group of $$\mathbb Z/4$$.
Then $$\mathcal A$$ is spanned by $$\underline{X}_1,\underline{X}_2, \underline{X}_4$$ where $$\underline{X}_i$$ is the sum of all elements of order $$i$$.
Then $$\underline{X}_1=1$$ is the identity, $$\underline{X}_2^2 = 1$$, $$\underline{X}_2 \underline{X}_4 = \underline{X}_4$$, and $$\underline{X}_4^2 = 2 \underline{X}_2 + 2$$ so none can generate since $$\underline{X}_1=1$$ simply generates $$\mathbb Z$$, $$\underline{X}_2$$ generates the subspace spanned by $$1$$ and $$\underline{X}_2$$, and $$\underline{X}_4$$ generates the subspace spanned by $$1, \underline{X}_4, 2\underline{X}_2$$.