On normal subgroups which are direct products
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.
direct products of nonabelian simple groups
scientific article; zbMATH DE number 3955204