[Paper Review] The Number of Finite Groups Whose Element Orders is Given

A. R. Moghaddamfar, Wujie Shi|ArXiv.org|Sep 22, 2005

Finite Group Theory Research14 references3 citations

TL;DR

This paper investigates the recognition of finite groups by their element spectra, focusing on PGL(2, pⁿ) where p = 2^α3^β + 1 is prime. It proves that such groups are either uniquely recognizable (h(G) = 1) or nonrecognizable (h(G) = ∞), never almost recognizable. Key results show PGL(2,7) and PGL(2,9) are nonrecognizable, with explicit constructions of groups sharing their spectra.

ABSTRACT

The spectrum $ω(G)$ of a finite group $G$ is the set of element orders of $G$. If $Ω$ is a non-empty subset of the set of natural numbers, $h(Ω)$ stands for the number of isomorphism classes of finite groups $G$ with $ω(G)=Ω$ and put $h(G)=h(ω(G))$. We say that $G$ is recognizable (by spectrum $ω(G)$) if $h(G)=1$. The group $G$ is almost recognizable (resp. nonrecognizable) if $1

Motivation & Objective

• To determine the number of non-isomorphic finite groups sharing the same element spectrum (ω(G)) for PGL(2, pⁿ), where p = 2^α3^β + 1 is prime.
• To classify whether such groups are recognizable (h(G) = 1), almost recognizable (1 < h(G) < ∞), or nonrecognizable (h(G) = ∞).
• To prove that PGL(2,7) and PGL(2,9) are nonrecognizable by constructing explicit groups with identical spectra.
• To extend the understanding of Cpp-groups and their spectral properties in the context of projective linear groups.

Proposed method

• Utilizes the spectrum ω(G) — the set of element orders in a finite group G — and defines h(Ω) as the number of isomorphism classes of groups with spectrum Ω.
• Applies Zsigmondy’s Theorem to identify primitive prime divisors of qⁿ − 1 and qⁿ + 1, crucial for analyzing element orders.
• Employs group-theoretic techniques: analyzing centralizers, Sylow subgroups, and automorphism groups, particularly for L₂(q) and PGL(2,q).
• Uses the Cpp-group property (centralizer of any nontrivial p-element is a p-group) to restrict possible group structures.
• Constructs explicit extensions (e.g., of a 7-group by 2.S₄) to demonstrate spectral equivalence and non-uniqueness.
• Applies results from finite group theory, including the structure of Aut(L₂(q)) and the classification of simple Cpp-groups.

Experimental results

Research questions

• RQ1 For PGL(2, pⁿ) with p = 2^α3^β + 1 prime, is h(G) always 1 or ∞, and never in between?
• RQ2 Are PGL(2,7) and PGL(2,9) nonrecognizable, i.e., do they share their spectrum with other non-isomorphic groups?
• RQ3 Can primitive prime divisors of qⁿ − 1 be systematically computed to aid in spectral analysis?
• RQ4 What structural constraints (e.g., Cpp-property) force h(G) to be 1 or ∞ for PGL(2, pⁿ)?

Key findings

• PGL(2, pⁿ) with p = 2^α3^β + 1 prime are never almost recognizable; h(PGL(2, pⁿ)) is either 1 or ∞.
• PGL(2,7) is nonrecognizable, as there exists a Frobenius group extension of a 7-group by 2.S₄ with the same spectrum {6,7,8}.
• PGL(2,9) is nonrecognizable, confirmed via Lemma 7(2) and Lemma 4, which show multiple non-isomorphic groups share its spectrum.
• For PGL(2, pⁿ), if the Sylow 2-subgroup of the normal subgroup N is nontrivial, then h(G) = ∞, implying non-uniqueness.
• The spectrum of PGL(2, pⁿ) is µ(G) = {pⁿ − 1, p, pⁿ + 1}, and the presence of primitive prime divisors in these values determines recognition behavior.
• A computer program is developed to compute primitive prime divisors of aⁿ − 1, supporting spectral analysis in the proof.