Minimal classes and maximal class in \(p\)-groups
The authors study finite non-commutative \(p\)-groups \(G\) with the property that the number of minimal classes (i.e., the smallest non-singleton conjugacy classes) is \(p-1\). Among others, they prove the following results. For such a group, \(Z_2(G)\) is an elementary Abelian \(p\)-group, \(|Z_2(G):Z(G)|=p\) and the minimal classes are exactly the nontrivial cosets of \(Z(G)\) in \(Z_2(G)\). A metabelian group \(G\) has this property if and only if \(G\) is a CF-group (i.e., the lower central factors of \(G\), except the first one, have order \(p\)), the degree of commutativity of \(G\) is one, the nilpotence class of \(G\) is \(c\geq 4\) and \(C(G')\cap Z_{c-1}(G)=G'\) holds. A \(p\)-group of maximal class \(G\) of order \(p^n\) has the investigated property if and only if \([G_1,G_{n-3}]\neq 1\) (where \(G_i\) for \(i\geq 2\) denotes the \(i\)th term of the lower central series of \(G\) and \(G_1=C(G_2/G_4)\)). In addition, some results about metabelian CF-groups are proved as well, for example, that a metabelian \(p\)-group \(G\) of nilpotence class at least \(3\) is a CF-group if and only if \(|Z_2(G)\cap G'|=p^2\).
scientific article; zbMATH DE number 1310634
scientific article