{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T08:14:52Z","timestamp":1759133692628,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.\r\nThe 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 apartially 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 agroup if and only if $|G|$ and $|H|$ are coprime.\r\nThe paper concludes with a number of open problems.<\/jats:p>","DOI":"10.37236\/13413","type":"journal-article","created":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T13:57:02Z","timestamp":1745503022000},"source":"Crossref","is-referenced-by-count":1,"title":["On the Order Sequence of a Group"],"prefix":"10.37236","volume":"32","author":[{"given":"Peter J.","family":"Cameron","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hiranya Kishore","family":"Dey","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,4,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T13:57:02Z","timestamp":1745503022000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i2p9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,25]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,4,11]]}},"URL":"https:\/\/doi.org\/10.37236\/13413","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,4,25]]},"article-number":"P2.9"}}