Proving Group of Order p^2 is Cyclic or ZpXZp

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SUMMARY

In the discussion, it is established that if the order of a group G is p², where p is a prime number, then G is either cyclic or isomorphic to Zp × Zp. Key points include the existence of a non-trivial center in G, which leads to the conclusion that the center of G is the entire group, thereby confirming that G is abelian. The discussion emphasizes the application of theorems related to abelian groups to finalize the proof.

PREREQUISITES
  • Understanding of group theory concepts, specifically group order and cyclic groups.
  • Familiarity with the properties of abelian groups.
  • Knowledge of the center of a group and its significance in group structure.
  • Basic understanding of isomorphism in the context of group theory.
NEXT STEPS
  • Study the properties of cyclic groups and their classifications.
  • Learn about the structure and characteristics of abelian groups.
  • Explore the concept of the center of a group and its implications for group behavior.
  • Investigate theorems related to group isomorphisms, particularly in the context of finite groups.
USEFUL FOR

This discussion is beneficial for students and educators in abstract algebra, particularly those studying group theory, as well as mathematicians interested in the properties of finite groups.

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Homework Statement


If the order of G is p^2 and p is prime, then show that G is either cyclic or isomorphic to ZpXZp...



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The Attempt at a Solution


Any hints here will help!
 
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I think two helpful facts here is that if the |G|=p2 for prime p, then G has non-trivial Centre. Furthermore, the Normalizer of the group is greater than the Centre. You can use this to show that the Centre of G is the entire group G, which implies it is abelian. Then use some other theorems involving abelian groups to prove your theorem.
 

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