description/proof of that for group, symmetric subset, and positive natural number, subset to power of number is symmetric
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of symmetric subset of group.
- The reader knows a definition of finite product of subsets of group.
- The reader admits the proposition that for any group and any finite number of subsets, the inverse of the product of the subsets in any order is the product of the inverses of the subsets in the reverse order.
Target Context
- The reader will have a description and a proof of the proposition that for any group, any symmetric subset, and any positive natural number, the subset to the power of the number is symmetric.
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:
\(G\): \(\in \{\text{ the groups }\}\)
\(S\): \(\in \{\text{ the symmetric subsets of } G\}\)
\(n\): \(\in \mathbb{N} \setminus \{0\}\)
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Statements:
\(S^n \in \{\text{ the symmetric subsets of } G\}\)
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2: Note
This proposition cannot be immediately generalized to the case, \(S_{j_1} ... S_{j_n}\) where \(S_{j_1}, ..., S_{j_n}\) are symmetric, when \(G\) is not Abelian, because while \((S_{j_1} ... S_{j_n})^{-1} = {S_{j_n}}^{-1} ... {S_{j_1}}^{-1} = S_{j_n} ... S_{j_1}\) holds, "\(= S_{j_1} ... S_{j_n}\)" is not proved.
3: Proof
Whole Strategy: Step 1: apply the proposition that for any group and any finite number of subsets, the inverse of the product of the subsets in any order is the product of the inverses of the subsets in the reverse order.
Step 1:
\((S^n)^{-1} = (S^{-1})^n\), by the proposition that for any group and any finite number of subsets, the inverse of the product of the subsets in any order is the product of the inverses of the subsets in the reverse order.
But as \(S^{-1} = S\), \(= S^n\).
So, \(S^n\) is symmetric.
