I have the following group action question that I would like some advice on how should i proceed.
Let $X$ be the group $\mathbb{Z}/n\mathbb{Z}$ and $G$ be the group $(\mathbb{Z}/n\mathbb{Z})^*$. Let $G$ act on $X$ via
$$g\cdot x=gx$$
Given $m\in X$, show that the orbit containing $m$ is the same as the orbit containing $d$ where $(m,n)=d$.
I have tried using the following to solve it to no avail.
- Bezout's identity: There exists $a,b\in \mathbb{Z}$ such that $am+bn=d$
- Let $a,b>0$ be integers and $d = (a, b)$. Suppose $d|N$. Then the
linear congruence
$$ax \equiv N (\text{mod }b)$$
$\space\space\space\space\space\space\space$ has $d$ mutually incongruent solutions modulo $b$.
Does anybody have any advice on how I should solve this?
Best Answer
I think an approach based on the Chinese remainder theorem might work.
Let
$n = p_1^{q_1}... p_r^{q_r}$
be the prime factor decomposition for $n$. From the chinese remainder theorem, we have:
$\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/p_1^{q_1}\mathbb{Z}\times ... \times \mathbb{Z}/p_r^{q_r}\mathbb{Z}$.
Let $m = (m_1,...,m_r)$, and let $e=(e_1,...,e_r)$ such that
$e_i = gcd(p_i^{q_i},m_i)$.
Since $e_i$ divides $p_i^{q_i}$ in $\mathbb{Z}/p_i^{q_i}\mathbb{Z}$, it must be a power of $p_i$ (the largest one which divides $m_i$), and we have $m_i = k_i e_i$, where $k_i$ can't be divided by $p_i$ and so is invertible.
This implies that $m = ke$ where $k = (k_1,...,k_r)$ is invertible. We will now see that
$d= ge$, for some invertible $g$.
Let $d = (d_1,...,d_r)$. We must see that $d_i$ is equal to the largest power of $p_i$ that divides $m_i$ up to invertibles. Since $d$ divides $n$, $dl = n$ implies $d_il_i = 0$. (Correction by Explorer: there are two cases).
If $l_i\neq 0$, then $d_i$ is not prime with $p_i^{q_i}$ and so, it must be a prime power of exponent $\leq$ $e_i$ times invertibles. But due to Bezout's identity, $d= am = ake = ce$, which means that $d_i = c_i e_i$ and thus, $e_i$ divides $d_i$, and so, the exponents must be equal by Euclid's lemma.
If $l_i=0$, then $d_i$ is invertible and can't be divided by $p_i$, so neither can $m$, which implies that $e_i$ is invertible as well. Since both $e_i$ and $d_i$ are invertible, they differ by an invertible element $g_i= d_ie_i^{-1}$.