Simple Abelian Groups: Can They Be Classified?

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SUMMARY

Simple Abelian groups can be classified definitively as the cyclic groups of order p, where p is a prime number. This classification arises from the property that any subgroup of an Abelian group is normal, leading to the conclusion that a simple Abelian group can only have the trivial subgroup {e} and itself. This foundational result simplifies the understanding of simple Abelian groups in group theory.

PREREQUISITES
  • Understanding of group theory concepts, specifically Abelian groups.
  • Familiarity with the definition and properties of simple groups.
  • Knowledge of cyclic groups and their characteristics.
  • Basic understanding of prime numbers and their significance in mathematics.
NEXT STEPS
  • Research the properties of cyclic groups in more detail.
  • Explore the implications of normal subgroups in group theory.
  • Study the classification of simple groups beyond Abelian groups.
  • Investigate the role of prime numbers in group theory and their applications.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in the classification of groups and their properties will benefit from this discussion.

Dawson64
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I've been doing some work with simple Abelian groups and their generators, and I feel like there is a way to classify all of them, is this possible?
 
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Yep. It's been done. Since any subgroup of a abelian group is normal, a simple abelian group must have only {e} and itself as subgroups. Thus the only simple abelian groups are the (cyclic) groups of order p, where p is a prime.
 

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