Orbit vs. Cycle

Can someone explain to me the difference between an orbit and a cycle?

An orbit is a set, a cycle is a permutation of a set (which permutes its elements cyclically)

Reference: Jamie Mulholland http://www.sfu.ca/~jtmulhol/math302/notes/302notes.pdf

p. 25: Definition of n-cycle = cycle of order n.

Orbit vs. Cycle


p. 253-254: Definition of orbit.

Orbit vs. Cycle

Mariano Suárez-Alvarez already pointed out their difference, they are simply of different type, one is a set, the other one describes a permutation. But regarding your comment, and as nobody else said here, we have a close connection between cycles and orbits; maybe there is where your confusion comes from.

If we have a permutation $\sigma \in S_X$, then this is a cycle of length $k$ if there exists some set $S = \{ s_1, \ldots, s_k \} \subseteq X$ of size $k$ such that $\sigma(s_i) = s_{i+1}$ and $\sigma(s) = s$ for $s \in X \setminus S$. On the other side, given a group acting on some set $X$, then the orbit of some $\alpha \in X$ is $$ \alpha^G := \{ \alpha^g : g \in G \} $$ i.e. the set of points from $X$ the point $\alpha$ is mapped on by $G$; orbits could also be defined with respect to subgroups $H \le G$, then $\alpha^H := \{\alpha^h : h \in H \}$.

If we have some $g \in G$, then we can consider the orbits of the subgroup $\langle g \rangle = \{ g^i : i \in \mathbb N \}$ generated by $g$, $$ \alpha^{\langle g \rangle} = \{ \alpha^{g^i} : i \in \mathbb N \} $$ which corresponds to applying $g$ many times, for each $\alpha \in X$. And these orbits correspond to the cycle decomposition if we view $g$ as an element from $S_X$ (faithful group actions could be identified with subgroups of $S_X$). If $\langle g \rangle$ has just a single orbit, then it induces a cycle in $S_X$, conversely every cycle in $S_X$ corresponds to an element with just a single non-trivial orbit (where a trivial orbit is an orbit containing just one element, i.e. which is fixed), if we view that element as acting on $X$ in the way described above. We see that the orbit is the set $S$ in the definition of a cycle.

This little observation is often used without mentioning, some consequences are for example that if we have some $g \in G$, where $G$ acts on $X$ (or equivalently some $\sigma_g \in S_X$), then the cycle length's in its induced permutation on $X$ have length dividing the order of $g$, and hence $|G|$. This follows by basic theorems about group actions, in this case that the orbit lengths divide the index of the point stabilizer (orbit-stabilizer theorem), i.e. if $H = \langle g \rangle$ $$ |\alpha^H| = |H : H_{\alpha}| $$ where $H_{\alpha} = \{ h \in H: \alpha^h = \alpha \}$ is called the point stabilizer. Hence it divides $|H|$, the order of $g$.

On Wikipedia:Cyclic permutation there is also a good description, also check out group actions, as said there is a close connection. Hope this helps!