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# The Group Theory Concept: The Abelian Group

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Please address the following problem: A group of order p^2,where p is a prime number, is abelian.

https://brainmass.com/math/group-theory/group-theory-concept-abelian-group-65235

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A group of order ,where is a prime number, is abelian.

Solution:- Let be a group of order , that is .

Then

is a subgroup of

and

Hence
For is prime.

Suppose

Then

which imply that there exists such that .

is a proper subgroup of

For by assumption.

But is a subgroup of

...

#### Solution Summary

In this solution, a detailed, step-wise response has been constructed to illustrate how to conduct a proof to show that something is abelian. The solution is detailed and well presented in an attached Word document.

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