Counting groups of a given order

Groups of each order up to 100. Read an entry as row + column: row 60, column +4 is order 64, which has 267 groups.

Two patterns run through it:

• Every peak carries a high power of 2: 64 = 2⁶ (267), 96 = 2⁵·3 (231), 48 = 2⁴·3 and 80 = 2⁴·5 (52 each), 32 = 2⁵ (51), 72 = 2³·3² (50). No other order up to 100 reaches 17.
• The 1s are the cyclic orders. There is exactly one group of order n — the cyclic one — precisely when gcd(n, φ(n)) = 1. That covers every prime, and composites like 15, 33, 35 and 65 as well.

Counting Abelian Groups

Counting p-groups

Almost every small group is a 2-group. Of the 49,910,529,484 groups of order at most 2000, 49,487,365,422 — over 99% — have order 1024.

The curve bends upward: the count is not a power of the order. The Higman–Sims asymptotic formula gives it, for a prime p, as

f(pn)  =  p(2/27)n³ + O(n8/3),

so log f grows like n³ while log pⁿ grows like n — a straight line on these axes would mean a fixed power, and no fixed power fits. Higman (1960) proved the lower bound, Sims (1965) the upper, which is what fixes the leading coefficient at 2/27.

• OEIS A000679 — the sequence plotted above.
• Wikipedia: p-group

Counting Groups of "Square-Free" Orders

Other