definition of m-cycle on n-symmetric group
Topics
About: group
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of n-symmetric group.
Target Context
- The reader will have a definition of m-cycle on n-symmetric group.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( S_n\): \(= \text{ the } n \text{ -symmetric group }\)
\( m\): \(\in \mathbb{N} \setminus \{0\}\) such that \(m \le n\)
\( \{p_1, ..., p_m\}\): \(\subseteq S_n\) with any order of the elements
\(*(p_1, ..., p_m)\): \(\in S_n\)
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Conditions: \((p_1, ..., p_m): p_1 \mapsto p_2, ..., p_{m - 1} \mapsto p_m, p_m \mapsto p_1\) with every other element mapped to itself
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2: Natural Language Description
For any natural number, \(n \in \mathbb{N} \setminus \{0\}\), the \(n\)-symmetric group, \(S_n\), any natural number, \(m \in \mathbb{N} \setminus \{0\}\) such that \(m \le n\), and any subset, \(\{p_1, ..., p_m\} \subseteq S_n\), with any order of the elements, \((p_1, ..., p_m) \in S_n\) such that \((p_1, ..., p_m): p_1 \mapsto p_2, ..., p_{m - 1} \mapsto p_m, p_m \mapsto p_1\) with every other element mapped to itself
