How many finite groups are there?

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

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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 $2^{10}=1024$ . We see this by connecting the dots of certain families of groups.

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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 -groups.

How many nilpotent groups are there?

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

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

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

(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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Keywords:

enumeration, p-group, nilpotent group

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Cross-references: fixed, Hall-Witt identity, Jacobi identity, strong, p-groups, class, leading coefficient, positive, factor, Frattini subgroup, class 2, lower bound, enumerating algebras, conjecture, simple, finite, bounds, proof, Sylow subgroups, product, nilpotent groups, distribution, square-free, integers, odd, interval, graph, order, groups, number, current

This is version 7 of enumerating groups, born on 2006-04-20, modified 2006-04-20.
Object id is 7849, canonical name is EnumeratingGroups.
Accessed 1240 times total.

Classification:

AMS MSC:

20F18 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Nilpotent groups)

Pending Errata and Addenda

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