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

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 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