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Subgroups with all finite lifts isomorphic are conjugate

This paper proves that non-conjugate subgroups of a finite group can be distinguished by an extension where their pre-images are non-isomorphic, thereby demonstrating that $\mathbb{Z}$ -coset equivalent subgroups are not necessarily isomorphic and resolving a question posed by Dipendra Prasad.

Original authors: Ido Karshon, Alexander Lubotzky, D. B. McReynolds, Alan W. Reid, Mark Shusterman

Published 2026-05-06

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CC BY 4.0

Original authors: Ido Karshon, Alexander Lubotzky, D. B. McReynolds, Alan W. Reid, Mark Shusterman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a large, complex machine (let's call it Group G). Inside this machine, there are two smaller, specific parts or sub-assemblies (let's call them Subgroup 1 and Subgroup 2).

The central question of this paper is: If these two sub-assemblies look different from each other inside the main machine, can we build a bigger, more complex machine that contains them, where they still look different?

Usually, mathematicians know that if two parts are "conjugate" (a fancy way of saying they are identical twins that have just been rotated or shifted to a different spot), they will always look the same no matter how you expand the machine. But what if they are not identical twins? What if they are genuinely different shapes?

The Main Discovery: The "Identity Card" Test

The authors, Karshon, Lubotzky, and their team, proved a powerful rule: If two subgroups are not identical (non-conjugate), there is always a way to build a larger, finite machine (an "extension") where their "blueprints" (pre-images) are clearly different.

Think of it like this:

You have two different keys, Key A and Key B. They look different.
You put them both into a standard lock (Group G).
The paper proves that you can design a special, larger lockbox (Group $\tilde{G}$ ) that contains the original lock.
When you try to fit the "lifted" versions of Key A and Key B into this new box, they will not be interchangeable. One might have a jagged edge the other doesn't, or a different number of teeth. They are fundamentally different in the new context.

The authors didn't just say this is possible; they showed how to build this new machine so that the "glue" holding it together (the kernel) is very well-behaved (supersolvable), ensuring the whole structure remains tidy and finite.

Why This Matters: The "Coset" Puzzle

The paper tackles a specific puzzle raised by mathematician Dipendra Prasad. In math, there's a concept called Z-coset equivalence.

Imagine Key A and Key B are different shapes.
However, if you look at how they interact with the rest of the machine (the "cosets"), they might produce the exact same pattern of noise or movement.
For a long time, people wondered: If two keys make the exact same noise pattern (are Z-coset equivalent), does that mean they must actually be the same shape?

The paper answers: NO.

Using their "larger machine" trick, the authors took a known pair of different keys (from a group called $PSL(2, 29)$) that made the same noise pattern. They built a bigger machine around them. In this new machine, the lifted versions of the keys were still making the same noise pattern, but their internal structures were now clearly different. This proves that making the same noise does not mean you are the same shape.

The "Magic Mirror" Analogy (Anabelian Geometry)

The paper also touches on a concept called Anabelian Geometry, which is like looking at a reflection in a magic mirror.

Imagine you have a shadow (a subgroup) cast on a wall.
In the world of "profinite groups" (infinite, fuzzy shadows), a famous theorem (Neukirch–Uchida) says that if two shadows look identical, they must be the same object.
The authors show that even if you can't see the whole infinite shadow, you can zoom in on a finite, clear picture (a finite group) and still tell the difference between two objects that look similar in the blurry background. It's like using a high-resolution camera to prove two blurry blobs are actually different animals.

The Computer Proof

To prove this works in the real world, the authors used a computer program (Magma) to build a specific example.

They took a specific group ($PSL(2, 29)$) with two different subgroups ( $A_5$ ).
They built a "parent" group ( $\Gamma$ ) that maps down to it.
They checked the "children" (the pre-images) of the two subgroups.
The Result: One child had 1 way to be broken down into smaller pieces of a certain size, while the other had 5 ways.
Since 1 is not equal to 5, the two children are definitely not the same shape, even though their parents looked similar in the original setup.

Summary

In simple terms, this paper says: If two groups are different, you can always find a bigger group where they remain different. You don't have to worry that they might accidentally become identical just because you added more context. This settles a specific question about whether "similar behavior" (coset equivalence) implies "identical identity," proving that it does not.

Technical Summary: Subgroups with All Finite Lifts Isomorphic Are Conjugate

Problem Statement
The paper addresses a fundamental question in group theory regarding the relationship between the isomorphism of subgroups and their behavior under group extensions. Specifically, it investigates whether the property of having isomorphic pre-images (lifts) in every finite extension of a group $G$ implies that the subgroups are conjugate within $G$ .

This inquiry is motivated by the study of $\mathbb{Z}$ -coset equivalence. Two subgroups $G_1, G_2 \leq G$ are $\mathbb{Z}$ -coset equivalent if the permutation modules $\mathbb{Z}[G/G_1]$ and $\mathbb{Z}[G/G_2]$ are isomorphic. While conjugate subgroups are trivially $\mathbb{Z}$ -coset equivalent, the converse is not generally true. A specific question raised by Dipendra Prasad (Question 1.3) asks: If $G$ is a finite group and $G_1, G_2 \leq G$ are $\mathbb{Z}$ -coset equivalent, are $G_1$ and $G_2$ necessarily isomorphic?

Prior to this work, very few examples of non-conjugate $\mathbb{Z}$ -coset equivalent pairs were known (notably a pair in $PSL(2, 29)$ found by L. Scott). The existence of such pairs where the subgroups are non-isomorphic was an open problem.

Methodology
The authors employ a constructive approach to generate finite group extensions that distinguish between non-conjugate subgroups. The core strategy relies on the following logical steps:

Reduction to Profinite Rigidity: The authors first establish that if two subgroups have non-isomorphic profinite completions, there exists a finite quotient where their pre-images are non-isomorphic (Proposition 3.1, Corollary 1.6).
Cornulier's Construction: The proof of the main theorem builds upon a result by Y. Cornulier (Theorem 1.2), which states that for any finite group $G$ , there exists a finite group $N$ such that $G \cong \text{Out}(N)$ .
Refinement of the Construction: The authors analyze and refine Sambale's exposition of Cornulier's construction (Theorem 4.1). They construct a specific finite group $N$ $N$ with the following critical properties (Theorem 4.2):
- $G \cong \text{Out}(N)$ .
- The center $Z(N)$ is trivial.
- The only subgroup of $\text{Aut}(N)$ isomorphic to $N$ is the inner automorphism group $\text{Inn}(N)$ .
- $N$ is supersolvable.
The Extension Group: Using the group $N$ , they define $\tilde{G} = \text{Aut}(N)$ . There is a natural surjective homomorphism $\phi: \tilde{G} \to G$ with kernel $N$ .
Contradiction Argument: The authors prove that if the pre-images $\phi^{-1}(G_1)$ and $\phi^{-1}(G_2)$ were isomorphic, one could construct an HNN extension embedding $\tilde{G}$ into a larger group where $G_1$ and $G_2$ become conjugate. By utilizing the specific rigidity properties of $N$ (specifically that $N$ is the unique subgroup of its isomorphism type in $\tilde{G}$ ), they demonstrate that this conjugacy in the extension forces $G_1$ and $G_2$ to be conjugate in the original group $G$ .

Key Results

Theorem 1.1 (Main Result): Let $G$ be a finite group. There exists a finite group $\tilde{G}$ and a surjective homomorphism $\phi: \tilde{G} \to G$ such that for any two non-conjugate subgroups $G_1, G_2 \leq G$ , the pre-images $\phi^{-1}(G_1)$ and $\phi^{-1}(G_2)$ are not isomorphic. Furthermore, the kernel of $\phi$ can be chosen to be supersolvable; if $G$ is solvable, $\tilde{G}$ is solvable.
Theorem 1.4 (Answer to Prasad's Question): The paper provides a negative answer to Question 1.3. By applying Theorem 1.1 to the known non-conjugate $\mathbb{Z}$ -coset equivalent pair in $G = PSL(2, 29)$ (where $G_1, G_2 \cong A_5$ ), the authors construct a finite group $\tilde{G}$ containing pre-images that are $\mathbb{Z}$ -coset equivalent but not isomorphic.
Theorem 1.7: The authors demonstrate a concrete application involving the group $\Gamma$ (a subgroup of index 2 in a reflection group of a tetrahedron). They show that $\Gamma$ has a finite quotient $Q$ mapping to $PSL(2, 29)$ such that the pre-images of the non-conjugate $A_5$ subgroups are non-isomorphic. This is verified computationally using Magma by showing the pre-images have different numbers of subgroups of index 5.

Significance and Connections
The paper claims significance in several areas of mathematics:

Resolution of an Open Question: It definitively answers Prasad's question, showing that $\mathbb{Z}$ -coset equivalence does not imply isomorphism of subgroups, even when the subgroups are non-conjugate.
Profinite Rigidity: The results connect to the subject of profinite rigidity. The paper suggests that if finite-index subgroups of certain groups (like mapping class groups or non-arithmetic lattices) are determined up to isomorphism by their profinite completions, then non-conjugate subgroups in those groups can be distinguished by finite quotients.
Anabelian Geometry: The work relates to the Neukirch–Uchida theorem. The authors note that while the theorem implies non-conjugate subgroups of absolute Galois groups are non-isomorphic, their result provides a finite-group analogue where the groups are finite rather than profinite.
Applications to Number Theory and Geometry: The existence of non-isomorphic $\mathbb{Z}$ $Z$ -coset equivalent pairs has implications for:
- Constructing non-isomorphic number fields with isomorphic multiplicative groups.
- Producing non-isomorphic Riemann surfaces with isomorphic Jacobians (without polarization).
- Studying the cohomology of spaces constructed from such pairs.

Modesty of Claims
The authors maintain a modest tone regarding the scope of their results. They explicitly state that while they provide a method to distinguish non-conjugate subgroups via finite lifts, the specific finite group $\tilde{G}$ constructed is often large and not necessarily minimal. They also note that while their construction works for all finite groups, the specific application to $\mathbb{Z}$ -coset equivalence relies on the existence of at least one initial non-conjugate pair (which was already known). The paper does not claim to classify all such pairs but rather establishes the existence of extensions that separate them.