Cycle notation

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In mathematics, cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles. This had sometimes been called circular notation and a permutation consisting of a single cycle was called a circular permutation. Modern terminology uses the term cyclic to mean a permutation with one cycle or one set of non-fixed points, and restricts a circular permutation to mean a permutation of objects arranged in a circle up to cyclic permutations, so that there is no fixed starting object on the circle.

Definition

Let $S$ be the set $\{1,\dots ,n\},n\in \mathbb {N}$, and

$a_{1},\ldots ,a_{k},\quad 1\leq k\leq n$

be distinct elements of $S$. The expression

$(a_{1}\ \ldots \ a_{k})$

denotes the cycle σ whose action is

$a_{1}\mapsto a_{2}\mapsto a_{3}\mapsto \ldots \mapsto a_{k}\mapsto a_{1}.$

For each index i,

$\sigma (a_{i})=a_{i+1},$

where $a_{k+1}$ is taken to mean $a_{1}$.

There are $k$ different expressions for the same cycle; the following all represent the same cycle:

$(a_{1}\ a_{2}\ a_{3}\ \ldots \ a_{k})=(a_{2}\ a_{3}\ \ldots \ a_{k}\ a_{1})=\cdots =(a_{k}\ a_{1}\ a_{2}\ \ldots \ a_{k-1}).\,$

A 1-element cycle such as (3) is the identity permutation. The identity permutation can also be written as the empty cycle, "()".

Permutation as product of cycles

Let $\pi$ be a permutation of $S$, and let

$S_{1},\ldots ,S_{k}\subset S,\quad k\in \mathbb {N}$

be the orbits of $\pi$. For an orbit $S_{j}$ $(j=1,\ldots ,k)$, let $n_{j}:=|S_{j}|$ denote the cardinality of $S_{j}$. Also, choose an element $a_{1,j}\in S_{j}$, and define

$a_{i+1,j}=\pi (a_{i,j}),\quad {\text{for }}1\leq i<n_{j};\quad {\text{then also }}\pi (a_{n_{j},j})=a_{1,j}.\,$

We can now express $\pi$ as a product of disjoint cycles, namely

$\pi =(a_{1,1}\ \ldots a_{n_{1},1})(a_{1,2}\ \ldots \ a_{n_{2},2})\ldots (a_{1,k}\ \ldots \ a_{n_{k},k}).\,$

In such an expression, it is typical, but not necessary, to suppress the 1-cycles. Thus, the permutation σ = (2 4 5)(1 6)(3) would be written as (2 4 5)(1 6) provided that it is understood that σ acts on S= {1,...,6}.

Since disjoint cycles commute with each other, the meaning of this expression is independent of the convention used for the order in products of permutations, namely whether the factors in such a product operate rightmost-first (as is usual more generally for function composition), or leftmost-first as some authors prefer. The meaning of individual cycles is also independent of this convention, namely always as described above.

Example

Here are the 24 elements of the symmetric group on $\{1,2,3,4\}$ expressed using the cycle notation, and grouped according to their conjugacy classes:

$()\,$

$(12),\;(13),\;(14),\;(23),\;(24),\;(34)$ (transpositions)

$(123),\;(132),\;(124),\;(142),\;(134),\;(143),\;(234),\;(243)$

$(12)(34),\;(13)(24),\;(14)(23)$

$(1234),\;(1243),\;(1324),\;(1342),\;(1423),\;(1432)$

See also

• Cyclic permutation
• Cycles and fixed points

References

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This article incorporates material from cycle notation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.