Existence, algorithms, and asymptotics of direct product decompositions. I. (Q2905577)

Let \(G\) be a finite group and fix a set \(\Omega\) of automorphisms of \(G\). We call \(H\) an \(\Omega\)-subgroup (respectively, \((\Omega\cup G)\)-subgroup) if \(H\) is a subgroup (respectively, normal subgroup) of \(G\) invariant under all \(\omega\in\Omega\). The object of this paper (and a paper to follow) is to present a constructive theory of direct decompositions of \(G\) into products of \(\Omega\)-subgroups, in particular, a decomposition into \(\Omega\)-indecomposables (called a Remak decomposition). A key idea is the concept of an \(\Omega\)-graded subgroup \(N\). This is a normal subgroup \(N\) such that if \(\mathcal H\) is the set of factors in a direct \(\Omega\)-decomposition of \(G\), then \(\{H\cap N\mid H\in\mathcal H\}\setminus\{1\}\) and \(\{HN/N\mid H\in\mathcal H\}\setminus\{N/N\}\) give direct (\(\Omega\cup G)\)-decompositions for \(N\) and \(G/N\), respectively.NEWLINENEWLINE Let \(\mathfrak X\) be a class of finite \(\Omega\)-groups closed under taking isomorphic images, direct \(\Omega\)-products and direct \(\Omega\)-factors. An up-\(\Omega\)-grader is an idempotent function \(G\mapsto\mathfrak X(G)\) of \(\mathfrak X\) into itself such that \(\mathfrak X(G)\) is \(\Omega\)-graded and for any direct \(\Omega\)-factor \(H\) of \(G\) we have \(\mathfrak X(H)=H\cap\mathfrak X(G)\). The idea is that starting from an \(\Omega\)-graded subgroup \(N\) we should be able to reconstruct a direct \(\Omega\)-decomposition of \(G\) from direct \((\Omega\cup G)\)-decompositions of \(N\) and \(G/N\).NEWLINENEWLINE Some typical theorems are the following. (Theorem 1): Every \(\Omega\)-group \(G\) has an \(\Omega\)-graded chief series. Thus either \(G\) is a characteristically simple \(\Omega\)-group and the Remak \(\Omega\)-decompositions are known (Theorem 6) or \(G\) has a proper nontrivial \(\Omega\)-graded subgroup \(N\). (Corollary to Theorem 2): an \(\Omega\)-group \(G\) is indecomposable if there is an up-\(\Omega \)-grader with \(\zeta_1(G)\leq\mathfrak X(G)\leq\Phi(G)\) for which \(G/\mathfrak X(G)\) is \(\Omega\)-indecomposable. (Theorem 3): If the \(\Omega \)-group \(G\) has an \(\Omega\)-grading pair \((\mathfrak X,G\mapsto\mathfrak X(G))\) and \(H=R\mathfrak X(G)\) for some direct \(\Omega\)-factor \(R\), then every direct \((\Omega\cup G)\)-factor of \(H\) either lies in \(\mathfrak X\) or is a direct \(\Omega\)-factor of \(G\). (Theorem 7): If \(\zeta_1(G)=1\) then \(G\) has a unique Remak \(\Omega\)-decomposition \(\mathcal R\) and the set \(\mathcal N\) of minimal \((\Omega\cup G)\)-subgroups is a direct decomposition of the socle of \(G\).