prove permutation is a cycle

**grad444** · September 15th 2008, 05:49 PM

Let p be a prime number. Prove that every power of a p-cycle is either a p-cycle or (1).

**ThePerfectHacker** · September 15th 2008, 06:12 PM

Originally Posted by grad444

Let p be a prime number. Prove that every power of a p-cycle is either a p-cycle or (1).

Let $\sigma$ be a $p$ -cycle. Then you can form the subgroup (of the symmetric group) $\left< \sigma\right>$ . Now $\sigma^n \in \left< \sigma \right>$ . Since $\left< \sigma \right>$ is a cyclic group of prime order it means the order of $\sigma^n$ is either $1$ or $p$ . If it is $1$ then $\sigma^n = (1)$ . Otherwise we know we can factor $\sigma^n = \tau_1\cdot ... \tau_k$ where $\tau_i$ 's are disjoint cycles, if $m_i$ is length of $\tau_i$ then $p=\text{ord}(\sigma^n) = \text{lcm}(m_1,...,m_k)$ . But since $m_i\not | p$ if $k>1$ (because it is prime) it means it is not possible for $\text{lcm}(m_1,...,m_k) = p$ . Therefore $k=1$ and so $\sigma^n$ must be a $p$ -cycle.

Thread: prove permutation is a cycle

LinkBack

Thread Tools

Display

prove permutation is a cycle

Similar Math Help Forum Discussions

prove S_p is generated by a p-cycle and an arbitrary transposition

Cycle type of non-disjoint permutation

cycle

cycle

any permutation times a k-cycle times the inverse is a k-cycle

Search Tags