Lifting property for finite groups

TL;DR

We address the problem of classifying finite groups whose mod $p$ representations lift to mod $p^2$ representations for all primes $p$. The authors develop an equivalence between lifting group homomorphisms to $GL_n(k)$ and lifting finite-dimensional $k[G]$-modules to $R[G]$-modules via an exact sequence with kernel $A_n(R)$, then analyze Sylow subgroups to constrain the global structure. They establish obstructions for several small groups (notably $C_2\times C_2$, $Q_8$, and $C_3\times C_3$) and prove liftability for cyclic groups of order $2^n$ and $3$, while showing non-liftability for cyclic groups of order $p^n$ with $p>2$. Combining these pieces yields a sharp classification: a finite group is liftable (and $p$-liftable for all primes) iff it is isomorphic to $C_{2^n}$, $C_3 \times C_{2^n}$, or $C_3 \rtimes C_{2^n}$. This result highlights the rigidity of the lifting property and connects it to module-theoretic and cohomological considerations within finite-group theory.

Abstract

We classify all finite groups that have lifting property of mod $p$ representations to mod $p^2$ representations for all prime $p$.

Theorems & Definitions (22)

• Theorem 1.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 3.1
• proof
• Corollary 3.2