Permutation cycle

A permutation cycle $(a_1{a}_2\dots a_k)$ sends $a_i$ to $a_{i+1}$, sends $a_k$ to $a_1$, and fixes every other element. Its order is $k$.

For $n>1$, every permutation of order of a permutation $n$ in $S_n$ is one full cycle if and only if $n$ is a prime power. If $n=p^r$, all lengths in its disjoint cycle decomposition divide $p^r$, and their least common multiple is their maximum. One cycle therefore has length $n$ and uses every symbol. If $n$ has more than one prime divisor, write $n=ab$ with $a,b>1$ coprime by separating one full prime-power factor from the others. Since $a+b\le ab=n$, disjoint cycles of lengths $a$ and $b$ fit inside $S_n$. Their product has order of a permutation $\mathrm{lcm}(a,b)=n$, but is not a full cycle.

A disjoint cycle decomposition writes a permutation as a product of disjoint permutation cycles. Repeatedly apply the permutation to a point until it returns to that point; these successive points form one cycle. Repeat on points not yet used. Finiteness and injectivity ensure that each process returns to its starting point. This partitions the underlying finite set into cycles. The decomposition is unique up to reordering the cycles, cyclically rotating the entries within each cycle, and optionally omitting fixed-point one-cycles. Include the one-cycles when specifying cycle type.

The cycle type of a permutation is the multiset of lengths in its decomposition into disjoint permutation cycles, including one-cycles for fixed points. Two permutations in the same symmetric group are conjugate exactly when they have the same cycle type.

Conjugation preserves order of a group element, because $(gx{g}^{-1}{)}^k=g{x}^k{g}^{-1}$, and preserves the sign of a permutation, because sign is a group homomorphism into $\{1,-1\}$. These invariants do not determine cycle type. In $S_6$, $(123)$ and $(123)(456)$ both have order three and positive sign, but have three and zero fixed points respectively. Conjugation relabels cycles, so these permutations are not conjugate.

Squaring an odd-length permutation cycle retains its length, whereas an even-length cycle splits into two cycles of half its length. Thus if any even cycle is present, the number of cycles strictly increases, so the cycle type changes. Since conjugate permutations in a symmetric group have the same cycle type, a permutation is conjugate to its square precisely when all its cycle lengths are odd, equivalently when its order of a group element is odd.

Left multiplication of a permutation by a transposition $(ab)$ swaps the two target labels $a,b$. If they occur in different cycles, $(a{x}_1\dots x_p)(b{y}_1\dots y_q)$ becomes the single cycle $(a{x}_1\dots x_{p}b{y}_1\dots y_q)$. If they occur in the same cycle, that formula reverses and splits the cycle in two. All other cycles are unchanged, so the total number of cycles, including fixed points, decreases or increases by exactly one. Empty intermediate lists yield one-cycles and require no exceptional case.

A three-cycle is a permutation cycle of length three. On any chosen set of three letters, the two possible three-cycles are inverses of one another.

Permutation cycles with disjoint supports commute. The order of their product is the least common multiple of their lengths.