On the order sequence of a group
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914104545902592 |
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| author | Cameron, Peter J. Dey, Hiranya Kishore |
| author_facet | Cameron, Peter J. Dey, Hiranya Kishore |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06516 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the order sequence of a group Cameron, Peter J. Dey, Hiranya Kishore Group Theory Combinatorics 20D15, 20D60, 20E22, 05E16 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. |
| title | On the order sequence of a group |
| topic | Group Theory Combinatorics 20D15, 20D60, 20E22, 05E16 |
| url | https://arxiv.org/abs/2310.06516 |