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 2im where m is odd and i large, for instance i>5. Indeed most groups are actually of order 210=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 groupsMathworldPlanetmath. 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 gnil(N) is the number of nilpotent groups of order N and g(N) the number of groups of order N then


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).

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 pn, denoted, f(pn), satisfies


for constants C1 and C2.

This result should be compared to the later work of Neretin on enumerating algebras. The lower boundMathworldPlanetmath is the work of Higman and is achieved by constructing a large family of class 2 p-groups (called Φ-class 2 groups as Φ(Φ(P))=1 where Φ is the Frattini subgroupMathworldPlanetmath of P).

The n8/3 factor has been improved to o(n5/2) by M. Newman and Seeley. Sims’ suggests that it should be possible to show


(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-groupsMathworldPlanetmathPlanetmath 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