My post is motivated at least in part by this MO question.
Has there been any work done on realizable order sequences for finite groups? By an "order sequence" I mean a non-decreasing list of the orders of the elements of the group. By "realizable" I mean there is some finite group that has that particular order sequence.
For example, $(1, 2, 4, 4)$ is a realizable order sequence; it is the order sequence of $\mathbb{Z}/4\mathbb{Z}$.
Is this something that has been studied in any depth? (Perhaps under a different name?) Are there any non-trivial theorems about realizable order sequences? (Examples of trivial theorems: ones that fall immediately out of Lagrange's Theorem; the degree sequence of $\mathbb{Z}/n\mathbb{Z}$)
I know that there are some nice theorems about degree sequences in graph theory (e.g., Erdős-Gallai theorem; Havel-Hakimi theorem), and though I have seen "order sequence" defined in a few Abstract Algebra texts, I have yet to come across any results of much interest.
I also wonder whether results such as this problem solution (Problem 6636, F. Schmidt, Amer. Math. Monthly, Vol. 98, No. 10 (Dec., 1991), pp. 970-972) or this paper (Isaacs et al (2009). Sums of element orders in finite groups. Commun. Alg. 37(9):2978-2980) contain ideas that would be applicable to such a topic.
Finally, is there an easily accessible (and organized) database that lists order sequences for all finite groups up to a certain not-too-small size?
Edit 1: Is anyone up to computing such a list and posting it somewhere accessible?
Edit 2: Now that Alexander Gruber has kindly posted computations for a fair number of order sequences, I wonder: (a) when does the same order sequence correspond to more than one group? (b) given an order sequence that corresponds to precisely one group, how difficult is it to recover the corresponding group's structure?
Edit 3: Mr. Gruber has most recently pointed me toward a related area of research on "OD-characterizability." One mathematician who has done a fair bit of work in this area is AR Moghaddamfar. See, for example, Recognizing Finite Groups Through Order and Degree Pattern.
Best Answer
Here's a quick list for groups up to order 512. Format is group order, followed by a list of possible order sequences for groups of that order, then an empty space, then a list of the number of groups with each respective order sequence (in the order they were listed). You can see it starts to get pretty long. Let me know if there is another specific range of orders you'd like me to run this code on.