Basics of Group Theory

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.