Ordering groups by element orders

Recently a couple of people have asked me questions, or suggested research topics, related to what appear to be elementary properties of finite groups, but on examination, show unexpected complexities. Part of the reason why group theory is such an endlessly fascinating topic. This one comes from Hiranya Kishore Dey in Bangalore, India. I hope to discuss the other one soon.

The paper we wrote is on the arXiv, 2310.06516. So you can read it if you want. There is an elephant in the room, which I will talk about later. (I hope this is an acceptable metaphor for a paper with an Indian co-author.)

Let G be a finite group of order n. Its order sequence, os(G), is the n-tuple of orders of the elements of G, arranged in non-decreasing order (so starting with 1 for the identity). The order sequences are partially ordered, where one sequence dominates another if every term of the first is at least as big as the corresponding term of the second. (Strictly speaking this is a partial order on sequences, not on groups, since there are pairs of groups with the same order sequence. It is a partial preorder on groups, but I will sometimes speak as if it were a partial order.)

There are two natural compositions on order sequences. One is the pointwise product, and the other is the pointwise least common multiple. (That is, if the ith elements of G and H are a and b respectively, then the ith element of their composition is ab for the product, lcm(a,b) for the lcm. These two operations coincide if the orders of the two groups are coprime. If they are not coprime, then the lcm gives the order sequence of the direct product, while the product is not realised by any group.

Moreover, if G and H have coprime orders, then the order sequence of their product dominates the order sequence of any extension of G by H.

There are parts of the poset of order sequences which have nice properties. For example, the order sequences of abelian groups of order p^m (for p prime) form a lattice isomorphic to the lattice of partitions of the integer m under the natural partial order on partitions, a very well studied lattice.

The property of nilpotency is characterised by the order sequence (in the sense that if a group is nilpotent then so is any other group with the same order sequence). If n is such that there exists a non-nilpotent group of order n, then there is a non-nilpotent group whose order sequence is dominated by the order sequence of every nilpotent group.

There are connections with graphs as well (of course). If two groups have isomorphic power graphs, then their order sequences are equal; and if their order sequences are equal, then their labelled Gruenberg–Kegel graphs are equal.

Now to that elephant. When Hiranya first contacted me, he asked me if I could prove that the order sequence of the cyclic group dominates the order sequence of any other group of the same order. If this were true, then a number of results in the literature would be easy consequences, while others could be substantially improved. Although this attractive conjecture is true for groups with order up to 1000, and the weaker statement that the order sequence of the cyclic group is maximal in the domination order is easily proved, we have been unable to show the general version. Help welcome!