Why $c(a_1 \ a_2 \dots \ a_k)c^{-1}=(c(a_1) c(a_2)… c(a_k))$? [duplicate]

cyclic-groups

Your question is about permutations and how conjugating a $k$-cycle by a permutation preserves the cycle type.

Claim: Let $c\in S_n$ and let $(a_1 \; a_2 \; \ldots \; a_k)$ be a $k$-cycle in $S_n$. Then $$ c (a_1 \; a_2 \; \ldots \; a_k) c^{-1} = (c(a_1) \; c(a_2) \; \ldots \; c(a_k))$$

Proof: Let's just consider how the left and right hand sides act on $c(a_1)$.

Then right hand side sends $c(a_1)$ to $c(a_2)$.

$c^{-1}$ sends $c(a_1)$ to $a_1$, which the $k$-cycle sends to $a_2$, which $c$ then sends to $c(a_2)$. Thus the left hand side as the composition of those three operations sends $c(a_1)$ to $c(a_2)$.

We chose $c(a_1)$ without loss of generality so we obtain the same agreement if we pick any $c(a_i)$. If a number in the set $\{ 1,\dots , n \}$ is not in the $k$-cycle, so it is not one of the $a_i$, then it is clearly fixed by both sides. So both sides are the same.

Related

Definite integral of a product of normal pdf and cdf
Showing $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}^2$
Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected.
Show that $f$ has at most one fixed point
Derivative of a Matrix with respect to a vector
Let $T:V→V$ be a linear transformation satisfying $T^2(v)=-v$ for all $v\in V$. How can we show that $n$ is even?
Proof of Aristarchus' Inequality
Proof of the universal property of the quotient topology
A scheme is affine iff the natural map $X\to \operatorname{Spec}\Gamma(X)$ is an isomorphism
Product Identity Multiple Angle or $\sin(nx)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(\frac{k\pi}n+x\right)$ [duplicate]
Should isometries be linear?
Prove $\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{2}\sqrt{\frac{\pi}{a}}\mathrm{e}^{-2\sqrt{ab}}$