Abstract Algebra/Group Theory/Group/Identity is Unique
Theorem
[edit | edit source]- Each group only has one identity
Proof
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As e1 is identity of G (usage 1), |
As e2 is identity of G (usage 1), |
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e2 is identity of G (usage 3), |
As e1 is identity of G (usage 3), |
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By 2a. and 3a., |
By 2b. and 3b., |
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By 4a. and 4b.,
- 5. , contradicting 1.
Since a right assumption can't lead to a wrong or contradicting conclusion, our assumption (1.) is false and identity of a group is unique.
Diagrams
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![]() as e2 is identity of G, and e1 is in G. |
![]() as e1 is identity of G, and e2 is in G |
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