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Number of groups of a given order?

  1. Feb 7, 2009 #1

    tgt

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    Is there a formula for determining the number of different groups up to isomorphism for a group of a given order?
     
  2. jcsd
  3. Feb 8, 2009 #2
    There isn't a general formula, but the number of groups has been tabulated for a large number of values.

    http://people.csse.uwa.edu.au/gordon/remote/cubcay/ [Broken] has a list of the number of groups up to order 1000. An interesting error causes it to say the number of groups of order 512 is -1, but it is actually 10,494,213.

    Mathematica 7 includes the function FiniteGroupCount, which will tell you the number of groups of a given order, up to 2047.
     
    Last edited by a moderator: May 4, 2017
  4. Feb 8, 2009 #3
    I should add: In general, it's quite hard to find the number of groups of a given order. Can you prove that there are exactly 5 groups of order 8? 5 groups of order 12? It's not trivial.

    Of course, for certain cases it's easy: Let p be a prime. Then there is exactly one group of order p (the cyclic one) and exactly two groups of order p2 (there are two abelian ones for sure, and it's a bit harder to show that every group of order p2 is abelian (hint: use the class equation)). It's harder to show that there are exactly five groups of order p3, but it's true.
     
    Last edited: Feb 8, 2009
  5. Feb 12, 2009 #4

    tgt

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    This seems such a fundamental question that more emphasis should be put on it. Is it a famous open problem as groups are so widely used.
     
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