I'm the one who thinks clear definition(clear with meta-language) is very important for doing mathematics.
Below, i list my definitions for cycle and orbit.
Let $X$ be a nonempty set.
Let $\langle \sigma \rangle$ be the cyclic subgroup generated by $\sigma$, where $\sigma\in S_X$.
Here, $\langle \sigma \rangle$ is a group acting on $X$ in a way that $\sigma^n . x = \sigma^n(x)$ for $x\in X$.
Then, orbit of $\sigma$ is $\langle \sigma \rangle . x$ for $x\in X$.
(This is consistent with the definition of orbit of a group action)
And below is the definition of cycle.
Let $\sigma$ be a permutation on a set $X$.
Then, $\sigma$ is a cycle iff there exists at most one orbit of $\sigma$ whose cardinality is greater than $1$.
With these definitions, how do one defines "disjoint cycles"?
Below is what i tried to formulate:
Let $\sigma,\tau$ be cycles on $X$.
Then, $\sigma,\tau$ are disjoint iff there does not exist $x\in X$, $| \langle \sigma \rangle . x |>1$ and $| \langle \tau \rangle . x|>1$.
Is this definition fine? Or if there is a clear definition of disjoint cycles please let me know. Thank you in advance:)
Best Answer
That formulation seems fine. What you've written is that $\sigma$ and $\tau$ are disjoint if nontrivial orbits of $\tau$ and $\sigma$ have no intersection.
Another way to think about this: let $X=\{1,\ldots,n\}$. We can express $\sigma=(i_1\,\cdots\,i_s)$ and $\tau=(j_1\,\cdots\, j_t)$, where $i_k,j_l\in X$ are numbers. Then $\sigma$ and $\tau$ are disjoint if and only if $i_k\neq j_l$ for all $k,l$.