Group number function

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

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

Prime power order

Further information: Enumeration of groups of prime power order

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

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 $1774274116992170$.

Fixed points of the group number function

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

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 $n<2047$.