We know that an identity permutation can be expressed as a product of zero transpositions. However, in some books, it is expressed as a product of non-zero transpositions. For example in the following permutation $f$.
$f=\begin{pmatrix} 1 ~2 ~3~ 4~ 5 ~6\\ 1~ 6~5~3~4~2 \end{pmatrix}=(1)(26)(354)=(12)(21)(26)(34)(35)$.
I do not understand why $(1)$ is expressed as $(12)(21)$? Specifically, why $2$ has been chosen here?
Will the following results also be true?
(a) $f=(13)(31)(26)(34)(35)$
(b) $f=(14)(41)(26)(34)(35)$
(c) $f=(15)(51)(26)(34)(35)$
(d) $f=(16)(61)(26)(34)(35)$, etc.?
As noted in the comment that all the expressions are correct. However, the use of a particular expression depends on the context of interest.
For instance a permutation $f=(1)$ is the same as
$f=\begin{pmatrix} 1 ~2 ~3~ 4~ 5 ~6\\ 1~ 2~3~4~5~6 \end{pmatrix}$.
Moreover, an identity permutation can be expressed as product of either zero transpositions or the product of a transposition with its inverse transposition.
These suffice that
(a) $f=(13)(31)(26)(34)(35)$
(b) $f=(14)(41)(26)(34)(35)$
(c) $f=(15)(51)(26)(34)(35)$
(d) $f=(16)(61)(26)(34)(35)$, etc. are all true.