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 qφ(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 pn 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.
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 qφ(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 pn 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.
| Original language | English |
|---|---|
| Article number | P2.9 |
| Number of pages | 21 |
| Journal | Electronic Journal of Combinatorics |
| Volume | 32 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 25 Apr 2025 |
Keywords
- Nilpotent group
- Order sequence
- Extension
- Partitions of integer