Pushing up Sp(4,2 k)
Let M be a finite group, p a prime and S a Sylow p-subgroup of M. Assume that no nontrivial characteristic subgroup of S is normal in M. Determine the structure of M. This problem is usually called the pushing-up problem for M. In his pioneering paper \textit{B. Baumann} [Proc. Am. Math. Soc. 74, 215-222 (1979; Zbl 0409.20009)] treated the case \(M/O_ 2(M)\cong L_ 2(2\) n). Independently \textit{R. Niles} [J. Algebra 57, 26-63 (1979; Zbl 0409.20007)] did the case \(M/O_ p(M)\cong SL_ 2(p^ n)\), p arbitrary. Using the so called amalgam-method \textit{B. Stellmacher} was able to give a very short treatment for \(M/O_ p(M)\cong SL_ 2(p^ n)\) [in Arch. Math. 46, 8-17 (1986; Zbl 0588.20013)]. In the paper under review the author treats the case \(M/O_ 2(M)\cong Sp(4,2\) n) using the amalgam-method. He proves the following: Let \((M/O_ 2(M))/\Phi (M/O_ 2(M))\cong Sp(4,2^ n)\), \(O(M)=1\). Set \(\tilde M=O^ 2(M)\) and \(V=O_ 2(\tilde M)\). Then one of the following holds: (i) \(\tilde M\cong Sp(4,2^ n)\), (ii) \(\tilde M/V\cong A_ 6\), V is the natural Sp(4,2)-module or the natural O(5,2)-module and \(\tilde M\) splits over V, (iii) \~M/V\(\cong 3A_ 6\) and V is the natural 6- dimensional module. Furthermore \(\tilde M\) splits over V, (iv) \(\tilde M/V\cong Sp(4,4)\), V is the natural Sp(4,4)-module and the extension is nonsplit.
scientific article; zbMATH DE number 4036053
scientific article