G=V4, S3, D4, Q8, A4 Find the following for each each above group 1) Cayley table 2) Inverse of each element 3) Order of group 4) Order of each element 5) Subgroups and Normal 6) center 7) conjugacy classes 8) Derived group 9) Factor group 10) Homomorphism 11) Automorphisms 12) Inner and outer

For each group G=V₄, S₃, D₄, Q₈, A₄: 1. **Cayley Table**: Describes the multiplication of all pairs of elements in the group. 2. **Inverse of Each Element**: Lists the inverse for every element in the group. 3. **Order of the Group**: The total number of elements in the group. 4. **Order of Each Element**: The number of times an element must be multiplied by itself to return to the identity element. 5. **Subgroups and Normal**: Identifies all subgroups and determines which are normal. 6. **Center**: Consists of elements that commute with every element in the group. 7. **Conjugacy Classes**: Groups elements that are conjugate to each other. 8. **Derived Group**: The subgroup generated by all commutators of the group. 9. **Factor Group**: The quotient of the group by its derived subgroup. 10. **Homomorphism**: Describes group homomorphisms between the groups. 11. **Automorphisms**: Automorphisms of the group. 12. **Inner and Outer Automorphisms**: Distinguishes between automorphisms that are inner (induced by conjugation) and those that are outer.