The embedding of CAP-subgroups in finite groups.
Let \(G\) be a finite group. A subgroup \(U\) of \(G\) is called a CAP-subgroup of \(G\) if each chief factor of \(G\) is either covered or avoided by \(U\). A subgroup \(U\) of \(G\) is called a subCAP-subgroup of \(G\) if there exists a chain of subgroups \(U=U_0\leq U_1\leq\cdots\leq U_r=G\) such that \(U_{i-1}\) is a CAP-subgroup of \(U_i\) for \(i=1,\dots,r\). The present paper is devoted to these subgroups and their influence on the structure of \(G\) and culminates in Proposition 28 stating that the following are equivalent: (1) \(G\) is soluble; (2) every Sylow subgroup of \(G\) is a CAP-subgroup of \(G\); (3) every maximal subgroup of \(G\) is a CAP-subgroup of \(G\); (4) if \(U\leq G\), then there exists a chain \(U=U_0\leq U_1\leq\cdots\leq U_r= G\) such that \(U_{i-1}\) is a CAP-subgroup of \(U_i\) and \([U_i:U_{i-1}]\) is a prime power for \(i=1,\dots,r\); (5) if \(U<G\), then there exists \(V\leq G\) such that \(U<V\) and \(U\) is a CAP-subgroup of \(V\); (6) every subgroup of \(G\) is a subCAP-subgroup of \(G\). -- Note that if, in Proposition 28, ``soluble'' is replaced by ``nilpotent'', ``CAP'' by ``normal'' and ``prime power'' by ``prime'', one obtains the classical characterisation of nilpotent groups.
scientific article; zbMATH DE number 5788846