# nt.number theory – Orbits of \$mathbb{Z}/nmathbb{Z}\$ under the action of its multiplicative group \$left(mathbb{Z}/nmathbb{Z}right)^{times}\$

Let $$mathbb{Z}/nmathbb{Z}$$, $$left(mathbb{Z}/nmathbb{Z}right)^{times}$$, and $$mathcal{D}_n$$ be the ring of integers modulo $$n$$, its multiplicative group of order $$n$$, and the set of divisors of $$n$$, respectively. The map $$phi: left(mathbb{Z}/nmathbb{Z}right)^{times}timesmathbb{Z}/nmathbb{Z}rightarrowmathbb{Z}/nmathbb{Z}$$, given by $$phi(g,x):=gx$$ is a left group action on $$mathbb{Z}/nmathbb{Z}$$. I am having a hard time in showing that $$phi$$ decomposes $$mathbb{Z}/nmathbb{Z}$$ into $$|mathcal{D}_n|$$-many distinct orbits given by $$left(mathbb{Z}/nmathbb{Z}right)^{times}cdotbar{delta}$$, where $$delta$$ is a divisor of $$n$$. Let me give an example. If $$n=12$$, then $$mathbb{Z}/12mathbb{Z}={bar{0},dots,bar{11}}$$, $$left(mathbb{Z}/12mathbb{Z}right)^{times}={bar{1}, bar{5},bar{7}, bar{11}}$$, and $$mathcal{D}_{12}={1,2,3,4,6,12}$$ and we have:
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{1}={bar{1}, bar{5},bar{7}, bar{11}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{2}={bar{2}, bar{10}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{3}={bar{3}, bar{9}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{4}={bar{4}, bar{8}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{6}={bar{6}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{0}={bar{0}},$$
and
$$bigcup_{delta|12}left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{delta}=mathbb{Z}/12mathbb{Z}.$$
I would be very grateful for any help or insights.

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