Math 2.0

Group theory and generalizations Definition

Cycle type

Builds on Decomposition of a permutation into disjoint cycles

Definition

Let σ∈Sn\sigma \in S_n. Write σ\sigma as a product of disjoint cycles, and include a 11-cycle for each fixed point. The cycle type of σ\sigma is the multiset of the lengths of these cycles.

Remarks

The disjoint cycle decomposition is unique up to order, so the cycle type is well defined. The lengths add up to nn, so the cycle type is a partition of nn. For example, (1 2)(3 4 5)∈S6(1\ 2)(3\ 4\ 5) \in S_6 has cycle type (3,2,1)(3, 2, 1).

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Cycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptSymmetric group, definition, no evidence levelSymmetric groupdefinitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementCycle type, definition, no evidence levelCycle typedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cycle notation concept
  2. Group concept
  3. Symmetric group definition
  4. Disjoint cycles commute statement
  5. Decomposition of a permutation into disjoint cycles statement
  6. Cycle type definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Permutation, Cycle type
About this page
Tag0BCHJ30REG
URIm2://0BCHJ30REG
Aliasescycle-structure
Kinddefinition
WritingW0 stub
MSC20B30
DefinesCycle type
UsesDecomposition of a permutation into disjoint cycles

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