SUMMARY
The discussion centers on whether A3 is a normal subgroup of S3. A3 is defined as H = {(1), (1 2 3), (1 3 2)}, while S3 is defined as G = {(1), (1 2 3), (1 3 2), (1 2), (1 3), (1 2 3)}. The conclusion drawn is that H is not a normal subgroup of G because the condition gH = Hg does not hold for all g in G, specifically demonstrated with g = (1 2 3), where gH ≠ Hg. The discussion also raises the question of the index [G:H] and its implications.
PREREQUISITES
- Understanding of group theory concepts, specifically normal subgroups.
- Familiarity with permutation groups, particularly S3.
- Knowledge of subgroup notation and operations.
- Basic understanding of the index of a subgroup.
NEXT STEPS
- Study the properties of normal subgroups in group theory.
- Explore the structure and properties of permutation groups, focusing on S3.
- Learn about the index of a subgroup and its significance in group theory.
- Investigate examples of normal and non-normal subgroups in various groups.
USEFUL FOR
Students and enthusiasts of abstract algebra, particularly those studying group theory, as well as mathematicians analyzing subgroup properties and their implications in permutation groups.
]?? What does that tell you??