On the normal complement problem for finite group algebras of Abelian-by-cyclic groups

TL;DR

This work addresses the normal complement problem for modular group algebras $FG$ where $G$ is an abelian-by-cyclic group $G \cong A \rtimes C_q$ with $A$ a finite $p$-group and $q$ an odd prime. The authors develop the unit-group structure, particularly the unitary and symmetric subgroups, and leverage conjugacy-class data to determine when a normal complement cannot exist in $V(FG)$; a key tool is the decomposition $V(FG) \cong (1+\Gamma(A)) \rtimes V(FB)$ and the analysis of $V(FB)$. The main result states that if $p^f = s q^{m} + 1$ with $(s,q)=1$ and either $m>1$ or $(s+1) \ge q$ with $2n \ge f(q-1)$, then no normal complement of $G$ exists in $V(FG)$ (and hence not in $\mathcal{U}(FG)$). This extends prior nonexistence results to the case $m=1$ and clarifies how the $q$-part of $|F^{\times}|$ and the action of $B$ influence the structure of unit groups in modular group algebras, with implications for understanding decompositions in these rings.

Abstract

Assume $F$ is a finite field of order $p^f$ and $q$ is an odd prime for which $p^f-1=sq^m$, where $m \ge 1$ and $(s,q)=1$. In this article, we obtain the order of symmetric and unitary subgroup of the semisimple group algebra $FC_q.$ Further, for the extension $G$ of $C_q = \langle b \rangle$ by an abelian group $A$ of order $p^n$ with $C_{A}(b) = \{e\}$, we prove that if $m>1,$ or $(s+1) \geq q$ and $2n \geq f(q-1)$, then $G$ does not have a normal complement in $V(FG)$.

Theorems & Definitions (23)

• Theorem 1.1
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Remark 3.1
• Corollary 3.2
• proof
• Lemma 3.3