Group number function

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Definition

The group number function or gnu function is the function $\mathrm{g}\mathrm{n}\mathrm{u}:\mathbb{N}\to \mathbb{N}$ defined by $\mathrm{g}\mathrm{n}\mathrm{u}(n)$ equal to the number of groups of order $n$ up to isomorphism.

Examples of values

• $\mathrm{g}\mathrm{n}\mathrm{u}(1)=1$.

Let $p$ be a prime number. Then:

• $\mathrm{g}\mathrm{n}\mathrm{u}(p)=1$, see classification of groups of prime order
• $\mathrm{g}\mathrm{n}\mathrm{u}(p^2)=2$, see classification of groups of prime-square order
• $\mathrm{g}\mathrm{n}\mathrm{u}(p^3)=5$, see classification of groups of prime-cube order
• $\mathrm{g}\mathrm{n}\mathrm{u}(2^4)=14$, $\mathrm{g}\mathrm{n}\mathrm{u}(p^4)=15$ for $p>2$, see Classification of groups of prime-fourth order
• $\mathrm{g}\mathrm{n}\mathrm{u}(2p)=2$, see classification of groups of order two times a prime

For $n$ a squarefree number, the value of $\mathrm{g}\mathrm{n}\mathrm{u}(n)$ is given by Hölder's formula:

$\mathrm{g}\mathrm{n}\mathrm{u}(n)={\sum }_{m\mid n}{\prod }_{p\mid n/m}\frac{p^{c(p,m)}-1}{p-1}$,

where $p$ is a prime, and $c(p,m)$ denotes the number of primes $q$ such that $q\mid m$, $q\equiv 1modp$.

Asymptotic bounds

A very weak bound for the number of groups of order $n$ up to isomorphism is $\mathrm{g}\mathrm{n}\mathrm{u}(n)\le n^{n^2}$, because this is simply the number of binary operations from a set to itself.

A better bound that can be proven using elementary methods is $\mathrm{g}\mathrm{n}\mathrm{u}(n)\le n^{n{\log }_{2}n}$. For full proof, refer: Number of groups of order n up to isomorphism is at most n to the power of (n log base 2 n)

Prime power order

Further information: Enumeration of groups of prime power order

Graham Higman demonstrated a bound for the group number function for groups of order $p^n$ for $p$ prime (i.e. p-groups), namely $p^{\frac{2}{27}{n}^2(n-6)}\le \mathrm{g}\mathrm{n}\mathrm{u}(p^n)\le p^{(\frac{2}{15}+{ϵ}_n)n^3}$ for some ${ϵ}_n\to 0$ as $n\to \mathrm{\infty }$.

Open problems

The following are currently open problems relating to the group number function.

Values of the group number function

Further information: Unclassified group orders

Certain values of the group number function are unknown, and thus the groups of that order are not classified. The smallest such example is for $\mathrm{g}\mathrm{n}\mathrm{u}(2048)$. See groups of order 2048. We do happen to know that the value of $\mathrm{g}\mathrm{n}\mathrm{u}(2048)$ strictly exceeds $1774274116992170$..

Non-trivial fixed points of the group number function

It is not known whether or not there is a number $n>1$ such that $\mathrm{g}\mathrm{n}\mathrm{u}(n)=n$.

Is every positive integer a group number? The gnu-hunting conjecture

For every $m\in \mathbb{N}$, does there exist $n\in \mathbb{N}$ such that $\mathrm{g}\mathrm{n}\mathrm{u}(n)=m$?

The galloping gnu conjecture

John H. Conway, Heiko Dietrich and E.A. O’Brien ask the question: does, for every $n$, the sequence $n,\mathrm{g}\mathrm{n}\mathrm{u}(n),\mathrm{g}\mathrm{n}\mathrm{u}(\mathrm{g}\mathrm{n}\mathrm{u}(n)),\dots$ eventually contain a $1$? They have verified it for $n\le 2047$.

In mathematical culture

The ID of the sequence of these numbers in the Online Encyclopedia of Integer Sequences is A000001

The values of the gnu function is the very first sequence in the OEIS, [1].

Table of values

See the page table of number of groups for small orders.

See also

• Minimal order attaining function