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

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 the symmetric and the 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 \((s+1) \ge q\) and \(2n \ge f(q-1)\), then G does not have a normal complement in V(FG).