In this link here it says that a cycle be decomposed a product of disjoint cycles. However say you consider the element of $S_4$ (The symmetric group of order 24) given by
$(12)(13)(14) = (1432)$. How would you write this as a product of disjoint cycles? An element in $S_4$ is either of the form $(1),(12),(123),(12)(43)$ or $(1234)$. So looking at the last cycle type we can never decompose it into a disjoint cycle unless we have something like $(1)(234)$. However if you decompose it into the form $(1)(234)$ or say $(14)(23)$ would this not contradict it being of the last cycle type?
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$\begingroup$$(1432)$ can be written as a product of disjoint cycles like so: $(1432)$. It's a product (with one factor) of disjoint cycles (each cycle in the product is disjoint from any other cycle in the product).
The main problem is that you misread the link. It doesn't say every cycle can be decomposed into a product of disjoint cycles, it says every permutation (on a finite set) can be decomposed into a product of disjoint cycles.
$\endgroup$ 3 $\begingroup$$(1432)$ is already a cycle, so there is nothing to decompose. In the decomposition of $(1432)$ into disjoint cycles there is only one term: the cycle $(1432)$ itself. The theorem about the decomposition of an arbitrary element of $S_n$ as a product of disjoint cycles allows the possibility that there is only one cycle in the decomposition.
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