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

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

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).

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

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

$$\frac{2}{27}{n}^{3}+{C}_{1}{n}^{2}\le {\mathrm{log}}_{p}f({p}^{n})\le \frac{2}{27}{n}^{3}+{C}_{2}{n}^{8/3}$$ |

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 and Seeley. Sims’ suggests that it should be possible to show

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

(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.

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