enumerating groups

1 How many finite groups are there?

The current tables list the number of groups up to order 2000 [Besche, Eick, O’Brien] (2000).

The graph is chaotic – both figuratively and mathematically.Most groups are distributed along the interval at values $2^{i}m$ where $m$ is odd and $i$ large, for instance $i>5$. Indeed most groups are actually of order $2^{10}=1024$. We see this by connecting the dots of certain families of groups.

Most integers are square-free, most groups are not [Mays 1980; Miller 1930;Balas 1966].

An explanation for this distribution is offered by considering nilpotent groups.Nilpotent groups are the product of their Sylow subgroups. So enumeratingnilpotent groups asks to enumerating $p$-groups.

2 How many nilpotent groups are there?

Theorem 1 (Pyber, 1993).

If $g_{n\mathit{}i\mathit{}l}\mathit{}\mathrm{(}N\mathrm{)}$ is the number of nilpotent groups of order $\mathrm{\le }N$ and $g\mathit{}\mathrm{(}N\mathrm{)}$ the numberof groups of order $\mathrm{\le }N$ then

$$\underset{N\to \mathrm{\infty }}{lim}\frac{\log g_{nil}(N)}{\log g(N)}=1.$$

The proof bounds the number of groups with a given set of Sylow subgroups and involvesthe Classification of Finite Simple Groups.

Conjecture 2 (Pyber, 1993).

$$\underset{N\to \mathrm{\infty }}{lim}\frac{g_{nil}(N)}{g(N)}=1.$$

If the conjecture is true, then most groups are 2-groups.

3 The Higman and Sims bounds

Theorem 3 (Higman 1960, Sims 1964).

The number of $p$-groups of order $p^n$, denoted, $f\mathit{}\mathrm{(}{p}^n\mathrm{)}$, satisfies

$$\frac{2}{27}{n}^3+C_1{n}^2\le {\log }_{p}f(p^n)\le \frac{2}{27}{n}^3+C_2{n}^{8/3}$$

for constants $C_{\mathrm{1}}$ and $C_{\mathrm{2}}$.

This result should be compared to the later work of Neretin on enumerating algebras. The lower bound is the work of Higman and is achieved by constructing a large family of class 2 $p$-groups (called $\mathrm{\Phi }$-class 2 groups as $\mathrm{\Phi }(\mathrm{\Phi }(P))=1$ where $\mathrm{\Phi }$ is the Frattini subgroup of $P$).

The $n^{8/3}$ factor has been improved to $o(n^{5/2})$ by M. Newman andSeeley. Sims’ suggests that it should be possible to show

$${\log }_{p}f(p,n)\in \frac{2}{27}{n}^3+O(n^2)$$

(with a positive leading coefficient) which would prove Pyber’s conjecture [Shalev].

S. R. Blackburn’s work (1992) on the number of class 3 p-groupsprovides strong evidence that this claim is true as he demonstrates that class 3 groups also attain this lower bound. Since class 3 groups involve the Jacobi identity (Hall-Witt identity) it is plausible to expect class c, for c less than some fixed bound, will asymptotically achieve the lower bound as well.