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My book gives this definition.

A permutation $z\in S_n$ is a transposition if:

  • there exist $i,j\in[n]=\{1,2,3,\dots,n\}$ with $i\ne j$, $z(i)=j$ and $z(j)=i$
  • for all $k\in[n]$ with $k\ne i$ and $k\ne j$, $z(k)=k$

According to this definition, is $(12)(34)(56)(7)$ a transposition? Because in his explanation he mentions that a "vast majority of cycles are singletons".

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Supoose $z = (1,2)(3,4)(5,6)(7)$.

$$z(1)=2, z(2)=1$$

Let $i=1$, $j=2$

However $z(3)=4$ even though $3 \neq 1$ and $3 \neq 2$.

Hence violating the second condition.

Hence $z$ is not a transposition.

Transposition switches exactly $2$ elements.

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  • $\begingroup$ Thx. I understood your point. $\endgroup$ – deepinlife Aug 21 '17 at 11:49

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