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Give the definition of an isometry and give examples.

State the group axioms.

State and verify the group axioms for the permutation group.

Give the definition of a subgroup.

How to determine a finite group is Abelian by its multiplication table?
Its multiplication table must be equal to its transpose.
Define a group isomorphism.

Given a multiplication table for a set equipped with a binary operation, how do we determine that it is a group?
First, note that if it is a group, every element appears exactly ONCE in each row or column.
Prove that the identity of element of a subgroup is the identity of the group.

State the subgroup criteria.

Prove that any subset of a group satisfying the subgroup criteria is a subgroup.

State and verify the orthogonal group as a subgroup of the Euclidean isometry group.

Define a group homomorphism.


Prove the above lemma for homomorphisms of groups.

Prove that the inverse of a group isomorphism is also an isomorphism.
