definition of n-alternating group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of n-symmetric group.
Target Context
- The reader will have a definition of n-alternating 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:
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( S\): \(= \{1, ..., n\}\)
\(*A_n\): \(= \{\text{ the even permutations on } S\}\) with the maps composition as the group operator, \(\in \{\text{ the subgroups of } S_n\}\)
//
Conditions:
//
2: Note
\(A_n\) is indeed a subgroup of \(S_n\): 0) for each elements, \(a_1, a_2 \in A_n\), \(a_1 a_2 \in A_n\), because as \(a_1\) and \(a_2\) can be realized as some sequences of even transpositions, \(a_1 a_2\) can be realized as the concatenation of the sequences, which is a sequence of even transpositions; 1) the associativity holds, because it holds on the ambient \(S_n\); 2) the identity map, \(id\), is in \(A_n\), because it can be realized as the sequence of 0 transposition; 3) for each element, the inverse element, which is the inverse map, is in \(A_n\), because it can be realized by the reverse sequence of the transpositions, which is a sequence of even transpositions.
