An elementary text on group theory applied to the Rubik's cube defines $S_n$, the symmetric group on n letters, as the set of bijections from $\{1,2,..., n\}$ to $\{1,2,...,n\}$ with the operation of composition.
A shorthand notation for such bijections is illustrated for some $\sigma\in S_{12}$: $$\sigma = (1\ 12\ 8\ 3\ 5\ 6\ 9\ 10)\ (2\ 4)\ (7)\ (11)$$
Some excerpts from the text:
Here, $(1\ 12\ 8\ 3\ 5\ 6\ 9\ 10)$, $(2\ 4)$, $(7)$, and $(10)$ are called cycles.
Definition 5.6. The cycle $(i_1\ i_2\ \dots\ i_k)$ is the element $\tau \in S_n$ defined by $τ(i_1)=i_2$, $\tau (i_2) = i_3$, ... , $\tau (i_{k−1}) = i_k$, $\tau (i_k) = i_1$ and $\tau (j) = j$ if $j \neq i_r$ for any $r$. The length of this cycle is $k$, and the support of the cycle is the set $\{i_1, \dots , i_k\}$ of numbers which appear in the cycle. The support is denoted by $\operatorname{supp}(\tau)$. A cycle of length $k$ is also called a k-cycle.
Definition 5.7. Two cycles $\sigma$ and $\tau$ are disjoint if they have no numbers in common; that is, $\operatorname{supp}{\sigma} \cap \operatorname{supp} \tau = \varnothing$.
My question: Is Definition 5.6 correct?
I find it problematic because it appears incompatible with the provided example $\sigma \in S_{12}$, where multiple "cycles" contain more than one number. Furthermore, the definition seems ambiguous since it refers to a cycle both as a bijection $\tau \in S_n$ as well as part of it ("The length of this cycle is $k$").