Extreme Reidemeister spectra of finite groups

This paper extends the concepts of the $R_\infty$-property and full Reidemeister spectrum to finite groups and identifies examples of groups exhibiting these properties by examining small-order and (quasi)simple groups.

The Gist

Imagine you have a massive, intricate dance floor filled with dancers. In the world of mathematics, this dance floor is a Group, and the dancers are the elements of that group.

Usually, when we study these groups, we look at how the dancers move around each other. But this paper introduces a special, slightly twisted way of dancing called "Twisted Conjugacy."

Here is the simple breakdown of what the author, Sam Tertoooy, is exploring, using everyday analogies.

1. The "Twisted" Dance Move

In a normal dance, if you want to move from position A to position B, you might just swap places with a friend. But in this "twisted" version, the rules change every time you move.

• The Rule: Two dancers are considered "twisted partners" if you can get from one to the other by doing a specific move involving a third dancer and a "twist" (an endomorphism, which is like a rulebook for how the dance changes).
• The Goal: Mathematicians want to count how many distinct "dance circles" (called Reidemeister classes) exist under these rules. This count is the Reidemeister Number.

2. The Two Extreme Worlds (Infinite vs. Finite)

For a long time, mathematicians only studied Infinite Groups (dance floors with an endless number of dancers). In those infinite worlds, there are two extreme scenarios:

1. The "R∞-Property": No matter how you change the rules, you always end up with an infinite number of dance circles.
2. The "Full Spectrum": You can create a dance rule that results in any number of circles you want (1 circle, 2 circles, 100 circles, infinity, etc.).

The Problem: You can't have these extremes in a Finite Group (a dance floor with a fixed, small number of dancers). If you only have 100 dancers, you can't have "infinite" circles, and you can't have every single number from 1 to infinity.

3. The New Definition: "Extreme" for Small Groups

The author asks: "If we can't have infinity, what does 'extreme' look like for a small, finite group?"

He redefines the extremes for finite groups based on the maximum possible number of circles, which is simply the total number of unique "conjugacy classes" (let's call this the Class Number, $k(G)$).

Extreme Case A: The "Boring" Group (Trivial Spectrum)

Imagine a group where, no matter how you change the dance rules, you always end up with the maximum possible number of circles.

• Analogy: It's like a group of dancers so rigid that no matter what song you play, they never change their formation. They always stay in their original, distinct spots.
• The Math: The only number of circles you can ever get is the maximum ($k(G)$).
• Who fits here? Groups with no "outer" changes allowed (like Symmetric groups $S_n$) or certain "perfect" groups where every rule change just shuffles the dancers within their existing circles.

Extreme Case B: The "Chameleon" Group (Full Extended Spectrum)

Imagine a group that is incredibly flexible. You can change the dance rules to get every single number of circles possible, from the absolute minimum (1 circle, where everyone is mixed together) to the absolute maximum.

• Analogy: This group is like a shape-shifting troupe. You can make them all merge into one giant blob, or split them into two, three, four, or up to the maximum number of distinct circles. They can do anything.
• The Math: The set of possible circle counts is $\{1, 2, 3, \dots, k(G)\}$.
• The Discovery: The author found that these groups are extremely rare. After checking thousands of small groups, he only found five that can do this:
  1. The trivial group (1 dancer).
  2. The group of order 2 (2 dancers).
  3. The Symmetric group $S_3$ (6 dancers).
  4. The Alternating group $A_4$ (12 dancers).
  5. A specific group of order 72 ($M_9$).

4. Why This Matters

The paper is essentially a "field guide" for these rare mathematical creatures.

• The "Odd Order" Rule: The author proves that if a group has an odd number of dancers, it can never be a "Chameleon." It will always be stuck with an odd number of circles. It's a mathematical law of physics for these groups.
• The "Quasisimple" Trap: Groups that look like simple groups (the building blocks of all groups) but have a tiny bit of "glue" (center) holding them together usually fail the "Chameleon" test. They are too rigid to achieve every number.

5. The Big Mystery

The author ends with a big question mark. He suspects that the five groups listed above are the only groups in the entire universe of finite groups that can achieve this "Full Extended Spectrum."

The Takeaway:
This paper takes a concept usually reserved for infinite, chaotic systems and applies it to small, finite systems. It discovers that while some groups are "boring" (always the same), and some are "flexible" (can do almost anything), the ones that can do everything are incredibly rare, like finding a unicorn in a stable. The author has found the five unicorns and is betting there are no others.

Technical Summary

1. Problem Statement and Context

The paper addresses the study of Reidemeister numbers ($R(\phi)$) and Reidemeister spectra ($\text{Spec}_R(G)$) within the context of finite groups.

• Background: In infinite group theory, twisted conjugacy classes are a major area of study. Two key "extreme" properties are well-known:
  1. The $R_\infty$-property: All automorphisms have an infinite Reidemeister number.
  2. Full Reidemeister spectrum: The set of Reidemeister numbers for all automorphisms includes every positive integer and infinity ($\mathbb{N} \cup \{\infty\}$).
• The Gap: These definitions rely on the existence of infinity, which is impossible for finite groups (where $R(\phi)$ is always finite). While twisted conjugacy on finite groups was studied by Fel'shtyn and Hill in the 1990s, it has only recently become a central object of study (e.g., by Senden).
• Objective: The author aims to define meaningful analogues of these "extreme" properties for finite groups, where the "infinity" bound is replaced by the class number $k(G)$ (the number of conjugacy classes), and to classify which finite groups satisfy these conditions.

2. Methodology

The paper employs a combination of theoretical group theory and computational algebra:

• Theoretical Framework: The author utilizes the Orbit-Stabilizer Theorem and properties of endomorphisms to establish bounds on Reidemeister numbers. Key tools include:
  • The relationship between $R(\phi)$ and the fixed points of the induced map on conjugacy classes.
  • The distinction between class-preserving automorphisms (those that map every conjugacy class to itself) and fixed-point-free endomorphisms.
  • Theorems regarding quasisimple groups and nilpotent groups.
• Computational Verification: Extensive computational searches were conducted using the GAP (Groups, Algorithms, Programming) system. Specific packages utilized include:
  • SmallGrp (for accessing groups of small order).
  • SmallClassNr (for class numbers).
  • TwistedConjugacy (for calculating Reidemeister numbers).
• Scope of Search: The author systematically analyzed groups of order $< 512$ for trivial spectra, groups of order $< 1536$ for full extended spectra, and groups with class numbers $< 15$.

3. Key Definitions and Theoretical Results

The paper establishes several foundational definitions and propositions:

• Trivial Reidemeister Spectrum: A finite group $G$ has a trivial spectrum if $\text{Spec}_R(G) = \{k(G)\}$.
  • Characterization: This occurs if and only if every automorphism of $G$ is class-preserving.
  • Examples: Groups with trivial outer automorphism groups ($\text{Out}(G)=1$), such as symmetric groups $S_n$ ($n \neq 6$), certain holomorphs, and specific $p$-groups.
• Trivial Extended Reidemeister Spectrum: A finite group $G$ has a trivial extended spectrum if $\text{ESpec}_R(G) = \{1, k(G)\}$.
  • Characterization: This occurs if and only if every endomorphism is either class-preserving or fixed-point-free.
  • Examples: Cyclic groups of prime order ($\mathbb{Z}_p$) and non-abelian finite simple groups (where $\text{Out}_c(G)=1$).
• Full Extended Reidemeister Spectrum: A finite group $G$ has a full extended spectrum if $\text{ESpec}_R(G) = \{1, 2, \dots, k(G)\}$.
  • Constraint: Unlike infinite groups, a finite group cannot have a full standard Reidemeister spectrum (i.e., it cannot achieve $k(G)-1$ via an automorphism).

4. Key Contributions and Results

A. Classification of Trivial Spectra

• The author provides a complete list of groups of order $< 512$ with trivial Reidemeister spectra (Table 1).
• It is shown that for finite simple groups, the spectrum is trivial if and only if $\text{Out}(G) = 1$. A comprehensive list of such simple groups (including specific Lie type groups, Mathieu groups, and the Monster group) is provided in Table 2.
• Open Question: Does there exist a finite group with trivial Reidemeister spectrum, trivial center ($Z(G)=1$), and non-trivial outer automorphism group ($\text{Out}(G) \neq 1$)?

B. Classification of Trivial Extended Spectra

• The paper identifies quasisimple groups (perfect groups with simple quotients) as a new source of examples.
• Proposition: If $G$ is quasisimple and $G/Z(G)$ has a trivial outer automorphism group, then $G$ has a trivial (extended) Reidemeister spectrum.
• Examples include $2.\text{Co}_1$, $2.\text{Ru}$, and $2.\text{B}$.
• Open Question: Are cyclic groups of prime order ($\mathbb{Z}_p, p \geq 3$) the only groups with trivial extended spectrum but non-trivial standard spectrum?

C. Classification of Full Extended Spectra

• Theorem 5.8: The author proves that the following families of finite groups cannot have a full extended Reidemeister spectrum:
  1. Groups of odd order.
  2. Nilpotent groups.
  3. Quasisimple groups.
  • Reasoning: These groups possess structural properties (e.g., non-trivial centers intersecting all normal subgroups) that prevent the existence of an endomorphism with $R(\phi) = k(G) - 1$.
• Computational Discovery: After searching groups of order $< 1536$, only five groups were found to possess a full extended Reidemeister spectrum:
  1. Trivial group ($[1,1]$)
  2. $\mathbb{Z}_2$ ($[2,1]$)
  3. $S_3$ ($[6,1]$)
  4. $A_4$ ($[12,3]$)
  5. $M_9$ (Mathieu group of order 72, $[72,41]$)
• Conjecture 5.9: The author conjectures that these five groups are the only finite groups with a full extended Reidemeister spectrum.

5. Significance

• Bridging Infinite and Finite Theory: The paper successfully translates concepts from infinite group theory (where "infinity" is the limit) to finite group theory (where the class number $k(G)$ is the limit), creating a coherent framework for "extreme" spectral properties in the finite realm.
• Computational Group Theory: It demonstrates the power of computational tools (GAP) in exploring structural properties of finite groups that are difficult to prove purely theoretically, leading to the identification of rare examples like $M_9$.
• New Research Directions: The paper raises several open questions regarding the existence of groups with specific combinations of center, outer automorphism groups, and spectral properties, guiding future research in the classification of finite groups via twisted conjugacy.
• Refinement of $p$-group Theory: It contributes to the ongoing study of $p$-groups and class-preserving automorphisms, a topic historically linked to questions posed by Mann.

In summary, this work provides the first systematic classification of finite groups based on the extremality of their Reidemeister spectra, offering both theoretical characterizations and a concrete list of examples that serve as a foundation for further investigation.