description/proof of memorandum on some multiplications of cycles on symmetric group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of n-symmetric group.
- The reader knows a definition of m-cycle on n-symmetric group.
Target Context
- The reader will have a description and a proof of a memorandum on some multiplications of cycles on any 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:
\(n\): \(\in \mathbb{N} \setminus \{0\}\)
\(S_n\): \(= \text{ the } n \text{ -symmetric group }\)
\(\{a, b, c, d\}\): \(\subseteq \{1, ..., n\}\)
//
Statements:
\((a, b)^2 = 1\)
\(\land\)
\((a, b, c)^2 = (a, c, b)\)
\(\land\)
\((a, b, c)^3 = (a, c, b) (a, b, c) = (a, b, c) (a, c, b) = 1\)
\(\land\)
\((a, b, c, d)^2 = (a, c) (b, d)\)
\(\land\)
\((a, b, c, d)^3 = (a, c) (b, d) (a, b, c, d) = (a, b, c, d) (a, c) (b, d) = (a, d, c, b)\)
\(\land\)
\((a, b, c, d)^4 = (a, d, c, b) (a, b, c, d) = (a, b, c, d) (a, d, c, b) = 1\)
\(\land\)
\((a, b, c, d) (a, b, d, c) = (a, c, b)\)
\(\land\)
\((a, b, c, d) (a, c, b, d) = (a, d, b)\)
\(\land\)
\((a, b, c, d) (a, c, d, b) = (a, d, c)\)
//
2: Note
Of course, the results are each ones that can be gotten by just doing diligently, but it is wasteful to do diligently every time, so, let us record them here as a reference for future occasions.
3: Proof
Whole Strategy: do them diligently.
\((a, b)^2 = (a, b) (a, b) = a \mapsto a, b \mapsto b = 1\).
\((a, b, c)^2 = (a, b, c) (a, b, c) = a \mapsto c, b \mapsto a, c \mapsto b = (a, c, b)\).
\((a, b, c)^3 = (a, b, c)^2 (a, b, c) = (a, c, b) (a, b, c) = a \mapsto a, b \mapsto b, c \mapsto c = 1\).
\((a, b, c)^3 = (a, b, c) (a, b, c)^2 = (a, b, c) (a, c, b) = a \mapsto a, b \mapsto b, c \mapsto c = 1\).
\((a, b, c, d)^2 = (a, b, c, d) (a, b, c, d) = a \mapsto c, b \mapsto d, c \mapsto a, d \mapsto b = (a, c) (b, d)\).
\((a, b, c, d)^3 = (a, b, c, d)^2 (a, b, c, d) = (a, c) (b, d) (a, b, c, d) = a \mapsto d, b \mapsto a, c \mapsto b, d \mapsto c = (a, d, c, b)\).
\((a, b, c, d)^3 = (a, b, c, d) (a, b, c, d)^2 = (a, b, c, d) (a, c) (b, d) = a \mapsto d, b \mapsto a, c \mapsto b, d \mapsto c = (a, d, c, b)\).
\((a, b, c, d)^4 = (a, b, c, d)^3 (a, b, c, d) = (a, d, c, b) (a, b, c, d) = a \mapsto a, b \mapsto b, c \mapsto c, d \mapsto d = 1\).
\((a, b, c, d)^4 = (a, b, c, d) (a, b, c, d)^3 = (a, b, c, d) (a, d, c, b) = a \mapsto a, b \mapsto b, c \mapsto c, d \mapsto d = 1\).
\((a, b, c, d) (a, b, d, c) = a \mapsto c, b \mapsto a, c \mapsto b, d \mapsto d = (a, c, b)\).
\((a, b, c, d) (a, c, b, d) = a \mapsto d, b \mapsto a, c \mapsto c, d \mapsto b = (a, d, b)\).
\((a, b, c, d) (a, c, d, b) = a \mapsto d, b \mapsto b, c \mapsto a, d \mapsto c = (a, d, c)\).
