Number of groups of a given order?

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Discussion Overview

The discussion revolves around the question of determining the number of different groups up to isomorphism for a group of a given order. It touches on theoretical aspects of group theory, including specific cases and challenges in counting groups of various orders.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about a formula for the number of groups of a given order, suggesting a general interest in the topic.
  • Another participant notes that while there is no general formula, the number of groups has been tabulated for many orders, providing a link to a resource that lists groups up to order 1000.
  • A specific error in the tabulated data is mentioned, where the number of groups of order 512 is incorrectly listed as -1, while the correct number is stated to be 10,494,213.
  • It is mentioned that Mathematica 7 includes a function, FiniteGroupCount, which can determine the number of groups of a given order up to 2047.
  • One participant emphasizes the difficulty of determining the number of groups for certain orders, specifically mentioning the challenge of proving the exact number of groups for orders 8 and 12.
  • Examples are provided regarding groups of prime order and their properties, including the assertion that there is one group of order p and two groups of order p², with a hint about using the class equation for further exploration.
  • A later reply suggests that the question of counting groups is fundamental and questions whether it is a famous open problem in the field.

Areas of Agreement / Disagreement

Participants generally agree that there is no simple formula for counting groups of a given order, and they acknowledge the complexity involved in determining the number of groups for specific orders. However, there is no consensus on the status of the problem as an open question.

Contextual Notes

The discussion highlights the limitations in current knowledge regarding the counting of groups, particularly for higher orders, and the dependence on specific mathematical tools and resources. Some assumptions about group properties and the nature of isomorphism are also implied but not explicitly stated.

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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?
 
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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/ 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.
 
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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.
 
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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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