# How to determine the number of isomorphism types of groups of a given order?

if $G$ is a group whose order is $n$ can we determine the number of isomorphism types for this number or not ?

for instance, if $n=4$ we have 2 types, $Z_4$ and $Z_2 \times Z_2$ " Klein 4-group"

for any number n, is a similar calculation possible ?

in other words, let $P$ is a function from Natural numbers into natural numbers which for any number $n$ gives the number of possible structures for a group of order $n$

can we find a formula for this function in terms of $n$ and using operation like addition, multiplication, etc ?

• No, it's not that easy. In fact, it can be extremely difficult. For abelian groups is easier and it is based on the name of partitions of the powers of the primes that divide $\,n\,$. – DonAntonio Jun 9 '13 at 21:13
• @DonAntonio , yup , it's easy for abelians groups using fundemental theorem for finite generated abelian groups or one of the other version of the theorem , but for nonabelian , is there no approximate answer ? or limited answer " e.g under particular conditions " ? – Fawzy Hegab Jun 9 '13 at 21:17
• to quote Aluffi's beautiful Algebra: Chapter 0 — To appreciate the difference in complexity, note that there are 42 abelian groups of order 1024 up to isomorphism... allegedly, there are 49,487,365,402 if we count noncommutative ones as well. – citedcorpse Jun 9 '13 at 21:17
• and then the footnote: This comparison is a little unfair, however, since it so happens that more than 99% of all groups of order < 2000 have order 1024. – citedcorpse Jun 9 '13 at 21:18
• I have written some related material in these answers: (1) (2) – Alexander Gruber Jun 10 '13 at 0:13

## 3 Answers

In view of all the information about how difficult and large $P(n)$ is, I should add the slightly consoling fact that it is algorithmically computable (and in fact primitive recursive). The reason this is only slightly consoling is that I can't think of an algorithm significantly better than a brute force search through all the groups, nested with brute force searches for isomorphisms between them.

• exactly, this very simple example shows just how far removed "computability" can be from feasibility – citedcorpse Jun 9 '13 at 21:24

There is a nice table on OEISWiki which shows the number of isomorphism classes for a group of order n - you should notice that they are quite sporadic. In particular, for groups of order $2^n$, the number of isomorphic classes grows quite considerably, especially relative to groups of similar size.

If you are seriously interested in this topic, you could look at the paper

Hans Ulrich Besche, Bettina Eick, and E.A. O'Brien. A millennium project: constructing small groups. Internat. J. Algebra Comput., 12:623-644, 2002,

which describes how the groups of order up to 2000 (which can be accessed in the GAP or Magma small groups library) were computed.

• is this paper a book or what ? edit: does this paper need a background from computational group theory ? – Fawzy Hegab Jun 10 '13 at 18:23