On the order sequence of a group

TL;DR

This work studies the order sequence $os(G)$ of a finite group and its domination relations, linking element orders to the power graph and Gruenberg–Kegel graph. It develops a framework of domination, strong domination, and product operations, and applies it to abelian $p$-groups via partitions to show a partition-lattice structure governing order sequences. The paper then derives sharp bounds on the product and sum of element orders, extends the analysis to wider classes of groups, and establishes tight equality conditions, while also revealing deep connections to group graphs. The results yield structural insights into nilpotent vs non-nilpotent behavior, prove partial orders on order sequences, and pose rich open questions about realizability and graph-theoretic relationships in finite groups.

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 $q^{φ(n)}$ times as large as the product in any non-cyclic group,where $q$ is the smallest prime divisor of $n$ and $φ$ 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.

Figures (2)

• Figure 1: Young diagrams
• Figure 2: Groups of order $60$

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 2.4
• Corollary 2.5
• Theorem 2.6
• Proposition 2.7
• Proposition 2.8