TITLE: Permutations cycles
QUESTION [0 upvotes]: Why is$$
        \begin{pmatrix}
        1 & 2 & 3 & 4 & 5 \\
        2 & 3 & 1 & 4 & 5\\
        \end{pmatrix}
$$
a cycle of length $3$, and
$$
        \begin{pmatrix}
        1 & 2  \\
        2 & 1 \\
        \end{pmatrix}
$$
a cycle of length $2$, but
$$
        \begin{pmatrix}
        1 & 2 & 3 & 4 & 5 \\
        2 & 1 & 4 & 5 & 3\\
        \end{pmatrix}
$$
is not a cycle at all?
Wouldn't the third permutation send $1$ to $2$ and $2$ back to $1$, completing the cycle of length $2$?

REPLY [2 votes]: The last one has two cycles, $(1\ 2)$ and $(3\ 4\ 5)$. It is not one singular cycle. In other words, it has cycles but it is not itself a cycle.
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