# 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 enumerating nilpotent groups asks to enumerating $p$-groups.

## 2 How many nilpotent groups are there?

###### Theorem 1 (Pyber, 1993).

If $g_{nil}(N)$ is the number of nilpotent groups of order $\leq N$ and $g(N)$ the number of groups of order $\leq N$ then

 $\lim_{N\rightarrow\infty}\frac{\log g_{nil}(N)}{\log g(N)}=1.$

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

###### Conjecture 2 (Pyber, 1993).
 $\lim_{N\rightarrow\infty}\frac{g_{nil}(N)}{g(N)}=1.$

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, $f(p^{n})$, satisfies

 $\frac{2}{27}n^{3}+C_{1}n^{2}\leq\log_{p}f(p^{n})\leq\frac{2}{27}n^{3}+C_{2}n^{% 8/3}$

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

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 $\Phi$-class 2 groups as $\Phi(\Phi(P))=1$ where $\Phi$ is the Frattini subgroup of $P$).

The $n^{8/3}$ factor has been improved to $o(n^{5/2})$ by M. Newman and Seeley. 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-groups provides 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.

Title enumerating groups
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