# How to know if a power is a permutation of an inverse group?

Consider the group $$ℤ^*_{55}$$

Is exponentiating to the 3rd power a permutation of: $$ℤ^*_{55}$$ And exponentiation to the 5th power?

I'm trying to solve this problem related to groups, but I don't know how to do it. Is there a mechanical way to find it? Something like a formula?

• Hint: Write the modulus ($n=55$ in the question) as a product of prime powers $\displaystyle n=\prod{p_i}^{k_i}$. Use the Chinese Remainder Theorem to reduce the problem to moduli of the form $n=\displaystyle{p_i}^{k_i}$. Solve that.
– fgrieu
Mar 23, 2022 at 18:36

A simpler approach is to prove that the map $$(\cdot)^p: \mathbb{Z}_m^{*} \rightarrow \mathbb{Z}_m^{*} \quad x \mapsto x^p$$ is a bijection. To do this determine the order of $$\mathbb{Z}_{m}^{*}$$, now by Lagrange's theorem you should know whether or not there are any elements of order $$p$$ (does $$p\mid\varphi(m)$$?). If there are elements of order $$p$$, can you see what the issue might be? If there are none, then injectivity follows with a short proof.
You might want to begin by considering an easier problem: for $$k, n \in \mathbb{N}$$, whether $$f(g)=g^k$$ is a permutation in the cyclic group $$C_n$$. Then you can look into the decomposition of $$\mathbb{Z}^\star_{55}$$ into cyclic groups and proceed from there.