# Number of abelian groups

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How many different non-isomorphic Abelian groups of order 16 are there?
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Is it 5?
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yes also please explain difference between isomorphic and non-isomorphic abelian groups
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I have encountered one such problem in this link

https://gateoverflow.in/1219/gate2007-21

There is a video attached in the comments about the isomorphism part but I am unable to understand it clearly

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@Mk Utkarsh

1)If we want non-isomorphic and non-abelian group, then what will be ans??

2) If we want isomorphic group of order 16, then what will be ans??

The number of Abelian groups of order  P^k (P is prime) is the number of partitions of  k.

Prime factorization of 16 = 2^4

The partitions of power i.e.., 4 = {1+1+1+1},{2+1+1},{2+2},{3+1},{4}

So total partitions =5

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Let G be a non abelian group, order of G can be 24 44 54 34
Question IF ( G, * ) is an abelian group then discuss correctness of each of the options ... a) a = a-1 for all a $\epsilon$ G b) a2 = a for all a $\epsilon$ G c) (a*b)2 = a2 * b2 for all a,b $\epsilon$ G d) G is finite order My ... b*a = a*b // cancelling a using left cancellation law and b using right cancellation law Therefore G is abelian group Given statement is correct D) PLEASE HELP ME ..