On normal subgroups which are direct products (Q1076798)

Let \(G\) be a group and \(M\) be a normal subgroup of \(G\). In this paper the authors consider the extension problem when \(G/M\) is a known group and \(M\) is a direct product of nonabelian simple groups. Assume that \(M\) is a direct product of simple groups \(S_ i\) such that \(\{S_ i\}\) is a conjugacy class of subgroups if \(G\). If \(M\) is a minimal normal subgroup of a finite group \(G\) and if \(M\) is nonabelian, then we have this situation. Let \(M=S\times K\) where \(S\) is a simple group, and set \(N=N_ G(K)\). Then, the group \(G\) is completely determined by the groups \(G/M\) and \(N/K\) together with an obvious homomorphism from \(N/K\) into \(G/M\) (Theorem 4.1). The authors prove that the extension of \(G\) over \(M\) splits if the extension of \(N/K\) over \(M/K\) splits. There is a one-to-one correspondence between conjugacy classes of complements to \(M\) in \(G\) and conjugacy classes of complements to \(M/K\) in \(N/K\). More generally, there is a bijection between conjugacy classes in \(N/K\) of subgroups \(L/K\) such that \(N/K=(M/K)(L/K)\) and conjugacy classes in \(G\) of subgroups \(H\) such that \(G=HM\) and \(H\cap M=\prod (H\cap M)/(H\cap K_ i)\) (the complete direct product) where \(\{K_ i\}\) is the set of all conjugate subgroups of \(K\) (Cor. 4.4 of Theorem 4.2). The construction of \(G\) from \(\alpha: N/K\to G/M\) is given by what the authors call the induced extension. Let \(\alpha\) be a homomorphism of a group \(A\) into a group \(B\). Let \(C=\alpha (A)\), let \(I\) be the set of right cosets of \(C\) in \(B\), let \(\rho\) be the permutation representation of \(B\) on \(I\), let \(P=\rho(B)\), and let \(T\) be a fixed right transversal of \(C\) in \(B\). For each \(x\in B\), let \(x_ T\) be the function defined on \(I\) by \(x_ T(Ct)=u_ T(tx^{-1})xt^{-1}\) where \(t\in T\) and \(u_ T(y)=Cy\cap T\). Let \(\lambda_ T\) be the mapping from \(B\) into the wreath product \(C Wr(P,I)\) defined by \(\lambda_ T(x)=\rho(x)x_ T\). It is known that \(\lambda_ T\) is a monomorphism from \(B\) into \(C Wr(P,I)\). Let \(G\) be the pull-back of the following diagram \[ A Wr(P,I)\to C Wr(P,I)@<\lambda_ T<<B \] where the first arrow is the map induced by \(\alpha:A\to\alpha(A)=C\). This group \(G\) is called the induced extension defined by \(\alpha\). Suppose that a normal subgroup \(M\) of a group \(G\) satisfies the following conditions: \(K\triangleleft M\), \(M\) is isomorphic to a complete direct product \(\prod (M/K_ i)\) where \(\{K_ i\}\) are the distinct conjugate subgroups of \(K\) in \(G\). Let \(N=N_ G(K)\). Then, \(G\) is isomorphic to the induced extension defined by \(\alpha:N/K\to G/M\) \((\alpha (Kx)=Mx)\). As an application of their theorems, they prove the following. Let \(B\) be any finite nonabelian simple group. Then, there is a finite group \(G\) with a minimal normal subgroup \(M\) such that \(M\) is the direct product of copies of the alternating group \(A_ 6\), \(G/B\) is isomorphic to \(B\), and \(G\) does not split over \(M\). (Clearly, there is a group \(G_ 0\) with the same normal structure such that \(G_ 0\) splits over \(M_ 0)\). The paper contains generalizations of the above results to the situation where \(\{K_ i\}\) is a union of conjugacy classes of subgroups. The paper ends with several interesting examples to illustrate why some of the theorems are the way they are.