Abstract Algebra/Group Theory/Group/Identity is Unique

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Theorem[edit]

Each group only has one identity

Proof[edit]

0. Let G be any group. Then G has an identity, say e1.
1. Assume G has a different identity e2

As e1 is identity of G (usage 1),

As e2 is identity of G (usage 1),

2a.
2b.

e2 is identity of G (usage 3),

As e1 is identity of G (usage 3),

3a.
3b.

By 2a. and 3a.,

By 2b. and 3b.,

4a.
4b.

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[edit]

1. Assume a group has two identities.
2. e1 * e2 = e1
as e2 is identity of G,
and e1 is in G.
3. e1 * e2 = e2
as e1 is identity of G,
and e2 is in G
4. The two identities are the same.
5. a group only has one identity