SUMMARY
Aut(S_3) is isomorphic to S_3, as both groups have exactly six elements. The group S_3, representing the symmetric group on three elements, is non-abelian, which distinguishes it from the cyclic group Z_6, also of order six. To prove the isomorphism, one must demonstrate that there are exactly six isomorphisms and identify at least one pair that do not commute. The relation (12)(13)=(132) is a key aspect in establishing the non-commutative nature of S_3.
PREREQUISITES
- Understanding of group theory concepts, specifically symmetric groups.
- Familiarity with isomorphisms and their properties in abstract algebra.
- Knowledge of permutation notation and operations.
- Basic understanding of abelian and non-abelian groups.
NEXT STEPS
- Study the properties of symmetric groups, focusing on S_3 and its structure.
- Learn how to construct and identify isomorphisms in group theory.
- Explore the differences between abelian and non-abelian groups.
- Investigate the concept of group actions and their implications in automorphisms.
USEFUL FOR
Students of abstract algebra, mathematicians interested in group theory, and anyone studying the properties of symmetric groups and their automorphisms.