permutation – Mathtuition88

S_n has Trivial Center for n greater than 3

We will prove that for $n\geq 3$ , $Z(S_n)=1$ .

Proof:

Clearly, $1\in Z(S_n)$ . Let $1\neq \sigma\in S_n$ . There exists $a, b$ distinct elements of $\{1, 2, \dots , n\}$ such that $\sigma(a)=b$ .

Consider the transposition $\tau =(b\ c)$ , where $c$ is distinct from $a, b$ . (Since $n\geq 3$ , such a $c$ exists.)

Then, $\tau\sigma (a)=\tau (b)=c$

$\sigma\tau (a)=\sigma (a)=b$

$\therefore \tau\sigma\neq\sigma\tau$

$\therefore \sigma \notin Z(S_n)$ .

Therefore, $Z(S_n)=1$

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Cycle Decomposition of Permutations is Unique

Cycle decomposition of Permutations into disjoint cycles is unique (up to reordering of cycles).

A proof can be found here.

The condition of disjoint is crucial. For example, the permutation (1 3 2) can be factored into (2 3)(1 2), where the two cycles are not disjoint. (1 3 2)=(1 2)(1 3) is also another decomposition, the two cycles are also not disjoint.

Wolframalpha can calculate permutations, a useful tool to replace manual calculations. Take note though that Wolframalpha’s convention is multiplying permutations from left to right, while most books follow the convention of multiplying right to left.

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