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In theorem 1: It is said that group is abelian if and only if order of group is prime number

In theorem 2 : It is said that the group having order as square of prime number is abelian

 

Theorem 1: https://yutsumura.com/a-simple-abelian-group-if-and-only-if-the-order-is-a-prime-number/

Theorem 2: https://gateoverflow.in/1066/gate2004-72 see the best answer

 

Both are contradicting 

 

If order of group is 4 :

then

Theorem 1: The group is not abelian as order is not prime.

Theorem 2: 4 = (2)^2  square of prime number hence abelian.

Please clarify ..

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You should read the first link carefully. It talks of a simple Abelian Group. Simple means no subgroup other than the trivial one and the group itself.
The second theorem does not state that the abelian group is simple.

So the first theorem should be:
A group is a Simple Abelian Group if and only if order of group is prime number.

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