SUMMARY
The discussion focuses on calculating the automorphism group Aut(Aut(Aut(C_{73}))). The conclusion reached is that Aut(Aut(Aut(C_{73}))) is isomorphic to Aut(C_2 × C_2 × C_2) × Aut(C_3), which further simplifies to GL_3(ℱ_2). The participants clarify that Aut(C_{73}) equals U(73), which is isomorphic to C_{72}, and that Aut(C_{72}) can be expressed as U(72) = U(2^3) × U(3^2) = C_2 × C_2 × C_2 × C_2 × C_3. A correction is made regarding the factors involved in the automorphism calculations.
PREREQUISITES
- Understanding of group theory concepts, specifically automorphism groups.
- Familiarity with the structure of cyclic groups, such as C_{73} and C_2.
- Knowledge of the unit group notation, U(n), and its properties.
- Basic understanding of linear algebra, particularly the general linear group GL_n over finite fields.
NEXT STEPS
- Study the properties of automorphism groups in group theory.
- Learn about the structure and properties of the unit group U(n) for various n.
- Research the general linear group GL_n(ℱ_q) and its applications in group theory.
- Explore the relationships between cyclic groups and their automorphism groups in detail.
USEFUL FOR
Mathematicians, particularly those specializing in group theory, algebraists, and students studying advanced topics in abstract algebra.