On the construction of matrix representations and their extensions
Let \(G=E\cdot H\) be the semi-direct product of finite groups E and H, (E is normal in G), with the following conditions: (1) E is a nonabelian p- group, p odd, and its center Z(E) is cyclic; (2) E/Z(E) is elementary abelian; (3) Z(E)\(\subseteq Z(E\cdot H)\); (4) H is a complement of Inn(E) in \(Aut_{Z(E)}(E)\). The main result of this paper is the following: If \(\psi\) is a faithful representation of E then an extension of \(\psi\) to G can be constructed. The author uses many steps to get to this aim. Some of these are studying the structure of E, deriving faithful representations of E, constructing a complement H of Inn(E) in \(Aut_{Z(E)}(A)\), constructing projective representations of \(E\cdot H\), and finally generalizing the results to arbitrary complements. In the last section of this paper, the author applies these results to construct certain characters of Sp(2m,q), \(m\geq 1\) and q an odd prime power.
scientific article; zbMATH DE number 4204639