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2009-05-02T20:26:22Z

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{{arithmetic function}}

==Definition==

Let <math>n</math> be a natural number. The '''number of groups''' of order <math>n</math> is the number of isomorphism classes of groups whose order is <math>n</math>. In other words, it is the maixmum possible number of pairwise non-isomorphic groups of order <math>n</math>.

==Behavior==

{{oeis|A000001}}

===Initial values===

For <math>n = 1,2,3,4,5,6,7,8,9,10,11,12</math> the number of groups is <math>1,1,1,2,1,2,1,5,2,2,1,5</math>.

===Lower bound===

The numbers <math>n</math> for which there is only one isomorphism class of groups of order <math>n</math> are termed [[cyclicity-forcing number]]s. These are also the only orders of groups whose automorphism group has coprime order to the group itself. {{further|[[Groupprops:Classification of cyclicity-forcing numbers]], [[Groupprops:Coprime automorphism group implies cyclic with order a cyclicity-forcing number]]}}

A number is a cyclicity-forcing number if and only if it is a product of distinct primes such that no <math>p_i</math> divides <math>p_j - 1</math> for prime divisors <math>p_i, p_j</math>. In particular, all prime numbers are cyclicity-forcing, and thus, the number of groups function takes the value <math>1</math> for infinitely many natural numbers. Note that there ''are'' composite cyclicity-forcing numbers, the smallest of which is <math>15</math>.

===Upper bound===

High values are typically attained at prime powers.

==Relation with other arithmetic functions==

* [[Number of nilpotent groups of given order]]: This is a [[multiplicative function]], unlike the total number of groups of given order.
* [[Number of abelian groups of given order]]: This is also a [[multiplicative function]], and is in fact the product of the [[number of unordered integer partitions]] for the exponents of each of the prime divisors of the order.

Number of groups of given order - Revision history

Vipul: Created page with '{{arithmetic function}} ==Definition== Let n be a natural number. The '''number of groups''' of order n is the number of isomorphism classes of groups...'

Vipul: Created page with '{{arithmetic function}} ==Definition== Let $n$ be a natural number. The '''number of groups''' of order $n$ is the number of isomorphism classes of groups...'