On the order sequence of a group

TL;DR

This paper explores the structure and properties of the order sequence poset of finite groups, revealing relationships with group types, prime factors, and partition posets, and establishing bounds and operations on these sequences.

Contribution

It introduces new results on the ordering and bounds of element orders in finite groups, linking order sequences to partition theory and group properties, and discusses conditions for sequence products to correspond to group orders.

Findings

The poset has a unique maximal element for cyclic groups.

Product of orders in cyclic groups exceeds that of non-cyclic groups by a factor related to prime divisors.

The order sequence poset of abelian p-groups is isomorphic to partition posets.

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. 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. 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. 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 $n$. 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.