description/proof of that for group, inverse of non-identity element is non-identity element
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 description and a proof of the proposition that for any group, the inverse of any non-identity element is a non-identity element.
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:
\(G\): \(\in \{\text{ the groups }\}\)
\(g\): \(\in G\)
//
Statements:
\(g \neq i\)
\(\implies\)
\(g^{- 1} \neq i\)
//
2: Proof
Whole Strategy: Step 1: suppose that \(g^{- 1} = i\) and find a contradiction.
Step 1:
Let us suppose that \(g^{- 1} = i\).
\(g g^{- 1} = i\), but the left hand side would be \(g i = g\), so, \(g = i\), a contradiction against that \(g \neq i\).
So, \(g^{- 1} \neq i\).
