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Every permutation of $n$ elements is a product of transpositions of the $n$ elements.

My work: We proceed by induction on $n$. Ovbiusly this stetement is true if $n=1,2$.
Now, suppose that $n\geq 3$ and the statement is true for all permutation on $n-1$ elements.

We need prove this is true for $n$ elements.

Let $\sigma\in S_n$.

Exists two cases, Suppose first that $\sigma(n)=n$. Then $\sigma |_{\{1,...,n-1\}}\in S_n$. is a product of transpositions by the inductive hypothesis, and then $\sigma$ is the product of the very same transpositions, regarded as permutations in $S_n$, fixing $n$.

Suppose next that $\sigma(n)\not = n$.

In this case i'm stuck. Can someone help me?

Moreover, if someone know other way to prove this please help me! Thanks.

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    $\begingroup$ What do you mean by transpositions of the $n$ elements? $\endgroup$ – GhD Dec 30 '17 at 17:42
  • $\begingroup$ They mean elements of $S_n$ that swap only two elements and fix the rest. $\endgroup$ – E-A Dec 30 '17 at 17:44
  • $\begingroup$ If you want to go on induction route, inducting on the sizes of cycles might be more helpful. Also, here is an answer that decomposes a cycle into transpositions. math.stackexchange.com/questions/319979/… $\endgroup$ – E-A Dec 30 '17 at 17:46
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If $\sigma(n)\neq n$, consider the transposition $\tau$ which exchanges $n$ and $\sigma(n)$. Then $\tau\sigma$ fixes $n$ and you can apply the induction hypothesis to it.

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Hint:

Suppose $\sigma(n)=a_1$, $\sigma(a_1)=a_2$ until $\sigma(a_i)=n$. If $i\leq n-2$ then every this permutation is product of disjoint permutation of length less than $n$ and by induction it is true you claim. For $i=n-1$ it is easy to find the transpositions!

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