What is a labeled group?

Question

Clicking through the OEIS I found this sequence that seems curious - the number of "labeled" groups. It cross references the usual sequence of the number of groups up to isomorphism (the very first sequence in the OEIS) as well as saying it is "a sequence related to groups".

So I've tried to look up this concept, but Google with "labeled group", "labeled group maths", "labeled group theory", and so on, to no avail at finding anything. Groupprops (essentially a wiki for group theory) doesn't seem to have anything either. I've clicked through the links on that OEIS entry and cannot find anything of help. So as a last resort I'm asking here (I know this isn't necessarily the kind of place for definition questions though, sorry).

So, what is a labeled group?

Answer 1

When classifying structures of a given type (groups, graphs, lattices, manifolds, etc.) one usually talks about isomorphism classes. So when one says "there are 2 groups of order 6", one actually means "up to isomorphism, there are 2 groups of order 6". Of course, literally, there are infinitely many groups of order 6 (in fact, there are infinitely many trivial groups!).

It turns out that the enumeration of all isomorphism classes is a very interesting and complex problem. However, it is much easier when one does not consider isomorphism classes but fixes an underlying set (often finite, when it comes to combinatorics).

For example, how many group structures are there on the finite set $\{1,2,3,4,5,6\}$? How many graphs are there with the finite vertex set $\{1,2,3\}$? These are called labelled objects (groups, graphs, etc.).

More generally, if $X$ is any set and $U : \mathcal{C} \to \mathbf{Set}$ is a functor, then one considers the set of $\mathcal{C}$-objects $A$ with $U(A)=X$. One says that $A$ has underlying set $X$, or equivalently, is labelled by $X$.

Answer 2

The sequence entry for OEIS sequence A034383 has a GAP program that leads to the answer to your question. The missing part of the sequence description is "up to isomorphism". That is, two labeled groups are considered to be equivalent if there is an isomorphism between them. In other words, given a fixed set of size $\,n\,$ (the labels) how many ways does it support a group structure provided that isomorphic groups are counted only once?

For example, for $\,n=4\,$ there are exactly two abstract groups of order $4$. The cyclic group and the Klein four group. The cyclic group has $2$ automorphisms. Thus the $\,4!=24\,$ ways to label this group splits into $\,24/2=12\,$ equivalence classes of size $2$. The Klein four group has $6$ automorphisms. Thus the ways to label this group splits into $\,24/6=4\,$ equivalence classes of size $6$. The total is $\,12 + 4 = 16\,$ so we get $\,a_4 = 16.\,$