On normal embedding of subgroups (Q1591224)

The author surveys conditions for a group to allow an embedding as a normal subgroup of another finite group, with an additional condition of containment in some given characteristic subgroup. The main objects of the article are finite groups. Let \(\Aut_c(G)\) denote the group of all central automorphisms of a group \(G\), \(\text{Inn}_c(G)=\Aut_c(G)\cap\text{Inn}(G)\) and let \(G_m\) be the \(m\)-th term of the lower central series of \(G\). Theorem 1. Let \(K\) be a finite \(p\)-group and \(L\) a Sylow \(p\)-subgroup of \(\Aut(K)\). If \(\text{Inn}(K)\subseteq\text{Inn}_c(K)L_m\) then there is a finite Abelian \(p\)-group \(F\) and a finite extension \(V\) of \(K\times F\) such that \(V_m\supseteq K\times F\). Theorem 2. Let \(K\) be a finite \(p\)-group with \(Z(K)\subseteq K'\). There is a finite \(p\)-group \(R\) with a normal subgroup \(N\cong K\) which is contained in \(R'\) if and only if the same condition holds for every direct factor of \(K\). The article contains also two results (Theorems 3 and 4) due to \textit{M.~Eyrich} [Diploma Thesis, Würzburg (1999)]. Theorem 3 states the following: if \(K\) is a finite \(p\)-group and \(\text{Inn}(K)\subseteq L_m\) for some Sylow \(p\)-subgroup \(L\) of \(\Aut(K)\) and \(m\geq 2\) then there is an extension \(V\) of the direct product \(D\) of \(p\) copies of \(K\) such that \(D\subseteq V_t\) where \(t=p(m-1)+1\).