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

Aleix Martí

7/23/23, 6:01 PM

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?

0 + 0

permutation

group-theory

fgrieu

7/23/23, 6:36 PM

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.

Score:0

Crypto

B.H.

7/23/23, 6:19 PM

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.

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Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: How to know if a power is a permutation of an inverse group

TH: จะรู้ได้อย่างไรว่ากำลังเป็นการเรียงสับเปลี่ยนของกลุ่มผกผัน

RO: Cum să știi dacă o putere este o permutare a unui grup invers

RU: Как узнать, является ли степень перестановкой инверсной группы

VI: Làm thế nào để biết nếu một sức mạnh là một hoán vị của một nhóm nghịch đảo

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