A class of soluble groups
A \(\lambda\) -group is a solvable finite group G having the property that -\(\alpha\) is a character of G, where \(\alpha (x)=\sup\{i: x\in G^{(i)}\}+1\) for \(x\in G\backslash\{1\}\) and \(\alpha (1)=-\sum\alpha (x)\), summed over \(x\neq 1\). The study of these groups is motivated by a result due to Artin who obtained essentially in this manner a character of the Galois group H of a finite Galois extension L/K from the normal series for H consisting of the ramification groups \(H_ i\). The author characterizes \(\lambda\) -groups (Theorem 2): A \(\lambda\) -group is (i) abelian, (ii) Frobenius of derived length 2, or (iii) a CN-group of derived length 3 with \(G/G'\cong C_ 2\), G'' a 2-group, G'/G'' cyclic of odd order, and both G/G'' and G' Frobenius. (Thus a \(\lambda\) -group is always a CN-group, i.e., the centralizer of each \(x\neq 1\) is nilpotent.) Furthermore each finite group of type (i), (ii) or (iii) is a \(\lambda\) - group. The proof is in three stages. First, necessary and sufficient conditions for G to be a \(\lambda\) -group are derived, in terms of the degrees of the irreducible characters of G. Then \(\lambda\) -groups of derived length 2 are classified with the consequence that no \(\lambda\) - group can have derived length 4 or more. Then a \(\lambda\) -group G of derived length 3 is classified by studying the action of G/G' on G''.
scientific article; zbMATH DE number 3875507