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Every permutation $p \in \sigma_n $ can be written as a product of disjoint cycles. For example, we can consider the following permutation $p \in \sigma_4 $ of the set of elements {1,2,3,4}: \begin{Bmatrix} 1 & 2& 3 & 4 \\ 2 & 3 & 4 & 1 \end{Bmatrix} that can be expressed as $(1 2 3 4)$ or \begin{Bmatrix} 1 & 2& 3 & 4 \\ 4 & 2 & 1 & 3 \end{Bmatrix} that can be expressed as $(1 4 3)\circ(2)$ In particular, every permutation $p \in \sigma_n $ can be written as a product of transpositions.

Every movement swaps two elements and, from this point of view, I see that every permutation is the product of a certain number of transpositions. But they can't be disjoint since the previous example $\begin{Bmatrix} 1 & 2& 3 & 4 \\ 2 & 3 & 4 & 1 \end{Bmatrix} =(1 2 3 4)$ can be written as $(1 2)(2 3)(3 4)(4 1)$

(Two cycles $(a_1,a_2,...,a_r)$ and $(b_1,b_2,...,b_s)$ are disjoint if $(a_1,a_2,...,a_r) \cap (b_1,b_2,...,b_s)=\varnothing $).

My question is how can I express the previous examples as a product of disjoint transpositions at the same time?

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    $(a_1,a_2,...,a_n)=(a_1a_n)(a_1a_{n-1})...(a_1a_3)(a_1a_2)$ – 2017-02-11
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    @Mustafa please can you be more clear? I'm a bit confused – 2017-02-11
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    any cycle is a product of transpositions, i.e, $(1234)=(14)(13)(12)$. and we have a corollary: any permutation of a finite set of at least two elements is a product of transposition. – 2017-02-11
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    @Mustafa can you suggest me a book where I can find all this things? – 2017-02-11
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    @Mustafa In the case that you showed the cycles are not disjoint – 2017-02-11
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    Name the book is: A first course in abstract algebra - Jb fraleigh, 7Ed, 2003, p90 – 2017-02-11
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    @Mustafa thank you very much – 2017-02-11

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You can write any cycle as the product of transpositions, but not necessarily disjoint transpositions, because in that case the order of the cycle will be 2, but the order of a k-cycle is $k$. You want to decompose in transpositions in order to compute the sign of a permutation, the fact transpositions are not disjoint is not a problem. For example $(123)$ cannot be the product of disjoint transpositions, but is $(12)(23)$ and so the sign is 1, this is a even permutation.