definition of 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
- 3: Note
Starting Context
- The reader knows a definition of group.
Target Context
- The reader will have a definition of 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:
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( S\): \(= \{1, ..., n\}\)
\(*S_n\): \(= \{\text{ the permutations on } S\}\) with the maps composition as the group operator
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Conditions:
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2: Natural Language Description
For any natural number, \(n \in \mathbb{N} \setminus \{0\}\), and the set, \(S := \{1, ..., n\}\), the set of the permutations on \(S\), \(S_n\), with the maps composition as the group operator
3: Note
\(S_n\) is indeed a group: 1) for any elements, \(p_1, p_2, p_3 \in S_n\), \((p_1 p_2) p_3 = p_1 (p_2 p_3)\), because \(p_j\) is a map and composition of maps is associative; 2) the identity map, \(id\), is the identity element; 3) for each element, the inverse map is the inverse element.
Of course, one can think of the permutations group of any other set of n-order, which is not exactly \(S_n\) but is obviously 'groups - homomorphisms' isomorphic to \(S_n\). We have specified \(S\) as \(\{1, ..., n\}\) in order to uniquely determine \(S_n\): very prevalently, the 'groups - homomorphisms' isomorphic groups are said to be "same", but we do no adopt that stance.
