SUMMARY
The discussion focuses on determining the number of non-isomorphic abelian groups of order 72, which is factored as 72 = 23 * 32. The groups identified include Z2 x Z2 x Z2 x Z3 x Z3, Z2 x Z2 x Z2 x Z9, Z2 x Z4 x Z3 x Z3, Z2 x Z4 x Z9, Z8 x Z3 x Z3, and Z8 x Z9. The fundamental theorem of finite abelian groups is crucial for this analysis, as it states that every finite abelian group can be expressed as a direct sum of cyclic subgroups of prime-power order.
PREREQUISITES
- Understanding of the fundamental theorem of finite abelian groups
- Knowledge of group theory concepts such as isomorphism and cyclic groups
- Familiarity with the prime factorization of integers
- Ability to compute least common multiples (LCM) of integers
NEXT STEPS
- Study the fundamental theorem of finite abelian groups in detail
- Learn about the classification of abelian groups by order
- Explore the concept of element order in direct products of groups
- Investigate examples of non-isomorphic groups and their properties
USEFUL FOR
Mathematics students, particularly those studying abstract algebra, group theory enthusiasts, and educators looking to deepen their understanding of abelian group classification.