Recovering a finite group's structure from the order of its elements.

Suppose you know the following two things about a group $G$ with $n$ elements:

1. the order of each of the $n$ elements in $G$;

2. $G$ is uniquely determined by the orders in (1).

Question: How difficult is it to recover the group structure of $G$? In other words, what is the best way to use this information to construct a Cayley table for $G$?

Note: (1) alone is not enough to uniquely determine a group. See this MO post for more.

Information about identifying when (1) implies (2) would be welcomed as well.

Solution 1:

This is actually quite a nontrivial question and is related to a concept called OD-characterizability, a topic of current research. Let me throw some definitions at you.

Definition. The prime graph of a group $G$ is a graph $\Gamma_G=\langle V, E \rangle$ where the vertex set $V$ is comprised of the prime divisors of $|G|$ and $\{p,q\}\in E$ if and only if there exists an element of order $pq$ in $G$. The degree pattern of a group $G$ is defined as $(\operatorname{deg}(p_1),\ldots,\operatorname{deg}(p_k))$ for $i=1,\ldots ,|V|$, where $\operatorname{deg}(p)$ denotes the degree of the vertex $p$ in $\Gamma_G$.

Definition. We say that a group $G$ is $n$-fold OD characterizable if there are exactly $n$ nonisomorphic finite groups with the same order and degree pattern as $G$. If a group $G$ is $1$-fold OD-characterizable, we simply say $G$ is OD-characterizable.

There is no reason why a group with a unique order sequence could not have the same degree pattern as another group of the same order. $p$-groups are an obvious example. On the other hand, assuming we know the order sequence of $G$, we can certainly construct the degree pattern of $G$. Of course, two groups with which have the same order sequence surely implies that they have the same degree pattern, so if a group is not uniquely determined by its order sequence it is not OD-characterizable. Thus every group which is OD-characterizable has a unique order sequence, and all the research which has been done about those should apply to your groups.

Unfortunately most OD-characterizability papers that I have seen focus on proving that certain classes of groups are OD-characterizable, e.g. alternating and symmetric groups, rather than on what OD-characterizability itself says about group structure. I suspect that's because it doesn't actually say a whole lot. For this reason I think that the best place to look if you planned to research this further would be at the order sequences of $p$-groups, as that is the primary place your condition differs from OD-characterizability.

However, not to be a bummer, but I wouldn't expect to be able to make any widesweeping statements. For example, amongst groups of order 32, there are $21$ order sequences. Out of those, the $10$ groups with unique order sequences are: $\mathbb{Z}_{32}$, $\left(\mathbb{Z}_{2}\right)^5$, $Q_{32}$, $D_{32}$, $D_{16}\times \mathbb{Z}_2$, $D_8 \times V$, the semidihedral group $SD_{32}$, the holomorph of $\mathbb{Z}_8$, and some nonabelian groups which are just referred to as $\text{SmallGroup}(32,7)$ and $\text{SmallGroup}(32,15)$. So whatever properties would be true of groups which are uniquely characterizable by their order sequences would have to be shared by all those groups, which as you can see are quite different.