definition of inverse of subset of group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of group.
Target Context
- The reader will have a definition of inverse of subset of 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:
\( G\): \(\in \{\text{ the groups }\}\)
\( S\): \(\subseteq G\)
\(*S^{-1}\): \(= \{g \in G \vert \exists s \in S (g = s^{-1})\}\)
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Conditions:
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2: Note
\(S^{-1} := \{g \in G \vert g^{-1} \in S\}\) is an equivalent definition.
Let us confirm that fact.
Let \(g \in \{g \in G \vert \exists s \in S (g = s^{-1})\}\) be any.
As \(g = s^{-1}\), \(g^{-1} = {s^{-1}}^{-1} = s \in S\), so, \(g \in \{g \in G \vert g^{-1} \in S\}\).
Let \(g \in \{g \in G \vert g^{-1} \in S\}\) be any.
Let \(s := g^{-1} \in S\).
Then, \(s^{-1} = {g^{-1}}^{-1} = g\).
So, \(g \in \{g \in G \vert \exists s \in S (g = s^{-1})\}\).
So, \(\{g \in G \vert \exists s \in S (g = s^{-1})\} = \{g \in G \vert g^{-1} \in S\}\).
When \(S\) is a subgroup of \(G\), \(S^{-1} = S\).
Let us see that fact.
Let \(g \in S^{-1}\) be any.
\(g^{-1} \in S\), but as \(S\) is a group, \(g = {g^{-1}}^{-1} \in S\).
Let \(s \in S\) be any.
As \(S\) is a group, \(s^{-1} \in S\), so, \(s \in S^{-1}\).
So, \(S^{-1} = S\).
But \(S^{-1} = S\) does not necessarily imply that \(S\) is a subgroup.
For example, let \(G = \mathbb{Z}\) be the additive group and \(S = \{-1, 0, 1\}\), then, \(S^{-1} = S\), but \(S\) is not any subgroup of \(G\): \(1 + 1 \notin S\).
