411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Group/Double Inverse
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Theorem
Let G be any group with operation .
In Group G, inverse of inverse of any element g is g.
Proof
{| style="width: 60%"
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| 0. Choose
||
|-
| 1.
|| definition of inverse of g in G (usage 1,3)
|-
| 2.
|| let a = g−1
|-
| 3.
||
|-
| 4.
|| definition of inverse of a in G (usage 2)
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| 5.
|| as a = g−1
|}
Diagrams
frame|1. inverse of filled circle is empty circle.frame|2. inverse of empty circle is filled circle, given 1.