Possible number of groups of order N

Question

When is the number of groups of some order $n$ greater than $n$? For example, lets say this happens at $n=3$, then that would mean that there are more groups of order 3 than 3.

$n=32$ is the smallest such $n$. There are $51$ isomorphism types of groups of order $32$. — Derek Holt, Nov 18 '14 at 21:32
I don't think there is any known $n>1$ for which there are exactly $n$ groups of order $n$. — Derek Holt, Nov 18 '14 at 21:37

score 0 · Answer 1 · edited Apr 13 '17 at 12:58

As already mentioned by Derek Holt in the comment, $n=32$ is the smallest $n$ such that there are more than $n$ groups of order $n$. This can be easily checked using the Small Groups Library which is contained in the computational algebra system GAP:

gap> First([1..100],n -> NrSmallGroups(n)>n);
32
gap> NrSmallGroups(32);
51
gap> List([1..32],NrSmallGroups);
[ 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 
1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51 ]

Regarding $n>1$ such that there are exactly $n$ groups of order $n$, this has been asked here and here, and no such $n$ is currently known.

Possible number of groups of order N

1 Answers1