The isomorphism problem for rational group algebras of finite metacyclic groups

TL;DR

The paper proves a positive instance of the Isomorphism Problem for group algebras in the rational setting: among finite metacyclic groups, the isomorphism type of the rational group algebra ${\mathbb Q}G$ determines the group itself. The authors develop and execute a program to recover the full MCINV$(G)=(m^G,n^G,s^G,\Delta^G)$ from ${\mathbb Q}G$, by showing that each invariant—$R^G$, $s^G$, $\epsilon^G$, $m^G$, $n^G$, $r^G$, and $\Delta^G$—is encoded in the Wedderburn decomposition via strong Shoda pairs and crossed-product components. They combine detailed prime-by-prime analyses of Sylow subgroups, cocyclic subgroups, and centers with Galois-theoretic arguments to demonstrate the reconstruction of MCINV$(G)$ from ${\mathbb Q}G$, and hence that two finite metacyclic groups with isomorphic rational group algebras must be isomorphic. This contributes to delineating the boundary between positive and negative isomorphism results for group algebras and reinforces the special role of metacyclic structure in the rational setting.

Abstract

We prove that if two finite metacyclic groups have isomorphic rational group algebras, then they are isomorphic. This contributes to understand where is the line separating positive and negative solutions to the Isomorphism Problem for group algebras.

Theorems & Definitions (50)

• Theorem A
• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5
• Theorem 2.6
• proof
• Example 2.7