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 where is odd and large, for instance . Indeed most groups are actually of order . We see this by connecting the dots of certain families of groups.
2 How many nilpotent groups are there?
Theorem 1 (Pyber, 1993).
If is the number of nilpotent groups of order and the number of groups of order 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 -groups of order , denoted, , satisfies
for constants and .
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 -groups (called -class 2 groups as where is the Frattini subgroup of ).
The factor has been improved to by M. Newman and Seeley. Sims’ suggests that it should be possible to show
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.