Question about permutations in $A_n$ A proof that $A_n$ is simple ($n>4$) begins as follows: Suppose $H$ is a nontrivial normal subgroup of $A_n$. We first prove that $H$ must contain a $3$-cycle. Let $\sigma \neq e$ a permutation that moves the leat number of integers in $n$, Being an even permutation $\sigma$ cannot be a cycle of even length. Hence, $\sigma$ must be a $3$-cycle or have a decomposition of the form $(a b c \cdots)\cdots$ or $(a b)(c d)\cdots$ , where $a,b,c,d$ are distinct. [CUT] > Why $\sigma$ cannot be, for example, $(abc)(def)$ ?

You are misinterpreting the notation. $(abc\cdot)\cdots$ means that the first cycle is of length _at least_ $3$, and there are possibly some other cycles afterwards. So $(abc)(def)$ _is_ contemplated in that notation, as is $(abcd)$.