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Lifting property for finite groups

This paper provides a complete classification of all finite groups that possess the lifting property, allowing their mod $p$ representations to be lifted to mod $p^2$ representations for every prime $p$ .

Original authors: Chandrashekhar B. Khare, Alexander Merkurjev

Published 2026-02-02

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CC BY 4.0

Original authors: Chandrashekhar B. Khare, Alexander Merkurjev

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 are trying to build a model of a complex machine using a specific type of Lego set. You have a "basic" set (let's call it Level 1) made of simple, flat bricks. You also have a "pro" set (Level 2) that includes the same bricks, but with a tiny, hidden layer of extra detail underneath them.

The paper by Khare and Merkurjev asks a very specific question about groups of symmetries (mathematical structures that describe how things can be rotated, flipped, or rearranged):

"If we can draw a perfect map of a symmetry group using our basic Level 1 bricks, can we always upgrade that map to the Level 2 pro set without breaking the rules?"

In the paper's language, this is called the "Lifting Property."

The Core Problem: The "Flat" vs. The "Deep"

Think of a mathematical group as a set of instructions for moving pieces around.

Level 1 (Mod $p$ ): This is a simplified view. It's like looking at a map where the terrain is perfectly flat. You can see the roads, but you can't see the hills or valleys.
Level 2 (Mod $p^2$ ): This is a slightly more detailed view. It's the same map, but now there's a tiny bit of depth (like a 1-millimeter elevation change) added to every point.

The authors are investigating: If a group works perfectly on the flat map, does it automatically work on the slightly deeper map?

The Big Discovery: Only "Simple" Groups Pass the Test

The authors found that for most groups, the answer is NO. If you try to upgrade the map, the instructions break. The "hills" in the new map cause the pieces to collide or get stuck in ways they didn't on the flat map.

However, they discovered that there are only three specific types of groups that are "Liftable" (meaning they can always be upgraded successfully):

The Simple Cyclic Group ( $C_{2^n}$ ): Imagine a single ring of dancers holding hands. They can rotate in a circle. This simple, single-loop structure is robust enough to handle the extra depth.
The Mixed Group ( $C_3 \times C_{2^n}$ ): Imagine a group of 3 dancers spinning in one direction, while a separate ring of $2^n$ dancers spins in another. As long as these two groups don't interfere with each other, they can be upgraded.
The Twisted Group ( $C_3 \rtimes C_{2^n}$ ): This is like the previous group, but the ring of $2^n$ dancers is "pushing" or "pulling" the group of 3 in a specific, non-trivial way. Surprisingly, this specific kind of interaction is stable enough to survive the upgrade.

The "Failures": Why Other Groups Break

The paper spends a lot of time proving why other famous groups fail this test. They use these groups as "counter-examples" to show how fragile the property is:

The Quaternion Group ( $Q_8$ ): Think of this as a 3D object with very specific, rigid rules for how it flips. The authors prove that if you try to add that tiny "depth" to the map, the rules for flipping contradict each other. It's like trying to fold a piece of paper that is too thick; it just won't crease correctly.
The Klein Group ( $C_2 \times C_2$ ): This is like a square where you can flip it horizontally or vertically. The authors show that when you add the extra depth, the horizontal flip and vertical flip start interfering with each other in a way that breaks the math.
Groups with too many 3s ( $C_3 \times C_3$ ): If you have two separate groups of 3 dancers, they create a grid that is too rigid to lift.

The "Sylow" Detective Work

To figure this out, the authors used a strategy like a detective breaking a crime scene into smaller clues.

They realized that if a whole group is "Liftable," then every small piece of it (specifically, the pieces made of a single prime number of elements, called Sylow subgroups) must also be Liftable.
They tested every possible small piece.
- Any piece with a prime number larger than 3? Fails.
- Any piece with 9 elements (three 3s)? Fails.
- Any piece with 4 elements arranged in a square? Fails.
- Any piece with 8 elements arranged like a quaternion? Fails.

By eliminating all the "bad" pieces, they were left with only the "good" pieces (cyclic groups of 2s and 3s). Then, they checked how these good pieces could be glued together. They found that only the three specific combinations listed above work.

The Bottom Line

The paper concludes that the "Lifting Property" is an incredibly strict filter. It's like a high-security gate that only lets through very specific, simple, or carefully balanced groups.

If you have a finite group of symmetries, it is almost certainly not liftable. It is only liftable if it is built from very specific, simple blocks (powers of 2) and perhaps a single block of 3, arranged in a very particular way. If your group is more complex (like the Quaternion group or a grid of 3s), the moment you try to add that extra layer of mathematical depth, the structure collapses.

Technical Summary: Lifting Property for Finite Groups

Problem Statement
This paper addresses the classification of finite groups $G$ possessing the "lifting property" for modular representations. Specifically, the authors investigate groups where every group homomorphism $G \to GL_n(k)$ (where $k$ is a field of characteristic $p$ ) extends to a homomorphism $G \to GL_n(R)$ for any commutative local ring $R$ with maximal ideal $M$ satisfying $M^2=0$ and residue field $k$ . Equivalently, this property requires that every finite-dimensional $k[G]$ -module lifts to a free $R[G]$ -module. The authors distinguish between a group being liftable (satisfying the condition for all such rings $R$ ) and $p$ -liftable (satisfying the condition specifically for $R = \mathbb{Z}/p^2\mathbb{Z}$ ).

Methodology
The authors employ a combination of cohomological arguments, module theory, and explicit matrix computations to establish necessary and sufficient conditions for lifting.

Reduction to Sylow Subgroups: Utilizing results from prior work (specifically [2]), the authors establish that a finite group $G$ is $p$ -liftable if and only if its Sylow $p$ -subgroup is $p$ -liftable. Furthermore, if every Sylow subgroup is liftable, the group itself is liftable. This reduces the global classification problem to analyzing $p$ -groups for each prime $p$ .
Obstruction via Ring Theory: For cyclic groups of order $p^n$ , the authors utilize the structure of the group ring $k[G]$ . They define an obstruction class $\theta(f, h)$ for elements $f, h \in k[G]$ such that $fh=0$. They prove that if $G$ is $p$ -liftable, then $\theta(f, h)$ must vanish for all such pairs. By constructing specific elements in cyclic groups of order $p^n$ (for $n \geq 2$ when $p=3$ , and $n \geq 2$ generally), they demonstrate non-vanishing obstructions, proving these groups are not $p$ -liftable.
Explicit Counterexamples: For the prime $p=2$ $p = 2$ , the authors perform explicit matrix calculations to disprove the lifting property for specific non-cyclic 2-groups:
- Klein Group ( $C_2 \times C_2$ ): They construct a 4-dimensional representation over $\mathbb{F}_2$ and show that the relations required for a lift to $\mathbb{Z}/4\mathbb{Z}$ lead to a contradiction involving the trace of matrices in $M_2(\mathbb{F}_2)$ .
- Quaternion Group ( $Q_8$ ): Similarly, they construct a 6-dimensional representation over $\mathbb{F}_2$ and demonstrate that the lifting relations imply a contradiction regarding the trace of commutators in $M_3(\mathbb{F}_2)$ .
- $C_3 \times C_3$ : They show that the group $C_3 \times C_3$ is not 3-liftable by analyzing the relations in $GL_3(\mathbb{Z}/9\mathbb{Z})$ for a specific representation, leading to a contradiction in the additive group of matrices over $\mathbb{F}_3$ .
Classification of $p$ -Liftable Groups:
- For $p > 3$ , the only $p$ -liftable $p$ -group is the trivial group, as any non-trivial $p$ -group contains a cyclic subgroup of order $p^2$ (or higher), which is not $p$ -liftable.
- For $p=3$ , the Sylow 3-subgroup must have order at most 3 (excluding $C_9$ and $C_3 \times C_3$ ).
- For $p=2$ , the Sylow 2-subgroup must be cyclic. The authors rule out generalized quaternion groups (which contain $Q_8$ ) and non-cyclic abelian groups (which contain $C_2 \times C_2$ ).

Key Results
The paper establishes the following equivalence for a finite group $G$ :

$G$ is liftable.
$G$ is $p$ -liftable for every prime $p$ .
$G$ $G$ is isomorphic to one of the following:
- $C_{2^n}$ (Cyclic group of order $2^n$ )
- $C_3 \times C_{2^n}$
- $C_3 \rtimes C_{2^n}$ (Semidirect product with respect to the unique non-trivial action of $C_{2^n}$ on $C_3$ ).

Significance and Claims
The authors position this work as a sharpening of previous results found in [2]. While [2] left open the question of whether the quaternion group $Q_8$ is liftable, this paper definitively proves that $Q_8$ is not 2-liftable. The paper claims that the lifting property is a "very restrictive" condition for finite groups.

The context provided relates to the modularity conjecture of Serre, where lifting mod $p$ representations of Galois groups to characteristic 0 is a central theme. While the lifting of representations for absolute Galois groups is constrained by the fact that only $1$ and $C_2$ are finite absolute Galois groups (by Artin-Schreier), this paper provides a complete classification for the dual problem: which finite groups allow the lifting of their modular representations. The authors note that their definitions of "liftable" and " $p$ -liftable" are shown to be equivalent to those in [2], thereby resolving ambiguities and closing open cases regarding specific non-abelian 2-groups.

The Core Problem: The "Flat" vs. The "Deep"

The Big Discovery: Only "Simple" Groups Pass the Test

The "Failures": Why Other Groups Break

The "Sylow" Detective Work

The Bottom Line

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