On the Order Sequence of a Group

Abstract

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group.

The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order $n$ are ordered by elementwise domination, forming a
partially ordered set. We prove a number of results about this poset, among them the following.
1. M. Amiri recently proved that the poset has a unique maximal element, corresponding to the cyclic group.
We show that the product of orders in a cyclic group of order $n$ is at least $q^{\phi(n)}$ times as large as the product in any non-cyclic group, where $q$ is the smallest prime divisor of $n$ and $\phi$ is Euler's function,
with a similar result for the sum.
2. The poset of order sequences of abelian groups of order $p^n$ is naturally isomorphic to the (well-studied) poset of partitions of $n$ with its natural partial order.
3. If there exists a non-nilpotent group of order $n$, then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order $n$.
4. There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups $G$ and $H$ is the order sequence of a
group if and only if $|G|$ and $|H|$ are coprime.

The paper concludes with a number of open problems.