Characterizing Metacyclic p-Groups with Identical Zeta Functions
Published:Dec 31, 2025 01:05
•1 min read
•ArXiv
Analysis
This paper addresses the problem of distinguishing finite groups based on their subgroup structure, a fundamental question in group theory. The group zeta function provides a way to encode information about the number of subgroups of a given order. The paper focuses on a specific class of groups, metacyclic p-groups of split type, and provides a concrete characterization of when two such groups have the same zeta function. This is significant because it contributes to the broader understanding of how group structure relates to its zeta function, a challenging problem with no general solution. The focus on a specific family of groups allows for a more detailed analysis and provides valuable insights.
Key Takeaways
- •The paper investigates the group zeta function as a tool for distinguishing finite groups.
- •It focuses on metacyclic p-groups of split type.
- •The main result provides a characterization of when two such groups have the same zeta function.
- •This contributes to the understanding of the relationship between group structure and its zeta function.
Reference
“For fixed $m$ and $n$, the paper characterizes the pairs of parameters $k_1,k_2$ for which $ζ_{G(p,m,n,k_1)}(s)=ζ_{G(p,m,n,k_2)}(s)$.”