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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:)

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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 $k,l$.

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  • $\begingroup$ Thank you, now i feel safe to use that definition in my post:) $\endgroup$
    – John. p
    Apr 17, 2014 at 23:42

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