# What is the definition of "disjoint cycles"?

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:)

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 $k,l$.