On the order sequence of a group
Peter J. Cameron, Hiranya Kishore Dey

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 is the list of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order 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 is at least times as large as the product in any non-cyclic group,where is the smallest prime divisor of and is Euler's function, with a similar result for the sum. The poset of order sequences of abelian groups of order is naturally…
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Taxonomy
Topicsgraph theory and CDMA systems · Coding theory and cryptography · Finite Group Theory Research
