# Number of groups of a given order?

• tgt
In summary, the number of groups of a given order can be determined using a formula based on the prime factorization of the order. This formula takes into account the number of unique prime factors and their powers to calculate the total number of distinct groups. Additionally, the number of groups can also be found by considering the divisors of the order and using the concept of cyclic groups. Overall, the number of groups of a given order depends on the prime factorization and the divisors of the order.
tgt
Is there a formula for determining the number of different groups up to isomorphism for a group of a given order?

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.

## 1. What does the "number of groups of a given order" refer to?

The number of groups of a given order refers to the number of unique mathematical structures that can be created with a certain number of elements and a particular operation.

## 2. How is the number of groups of a given order determined?

The number of groups of a given order is determined by using mathematical concepts such as group theory and combinatorics to analyze the possible combinations and arrangements of elements within a group.

## 3. What factors influence the number of groups of a given order?

The number of groups of a given order is influenced by the number of elements in the group, the type of operation being performed, and any restrictions or rules imposed on the group's structure.

## 4. Is there a limit to the number of groups of a given order?

There is no known limit to the number of groups of a given order. As the number of elements in a group increases, the number of possible groups also increases exponentially.

## 5. What is the significance of studying the number of groups of a given order?

Studying the number of groups of a given order can provide insight into the underlying patterns and structures of mathematics. It also has practical applications in fields such as cryptography, physics, and computer science.

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