\(p\)-groups of small breadth
The breadth \(b(G)\) of a finite \(p\)-group \(G\) is given by \(b(G)=\max_{x\in G}b(x)\), where \(p^{b(x)}=|G:C_G(x)|\). It is well-known that \(b(G)\leq 1\) is equivalent to \(|G'|\leq p\). A characterization of groups \(G\) satisfying \(b(G)=2\) and \(b(G)=3\) has been given just recently by \textit{N. Gavioli} et al. [J. Algebra 202, No. 1, 129-141 (1998; see the preceding review Zbl 0921.20017)] for odd \(p\) and \(p\geq 5\), respectively, using character theory. The authors investigate also just these groups but by purely elementary methods. At the same time they close the gaps except \(b(G)=3\) and \(p=2\), where only necessary conditions are derived. The case \(b(G)=3\) and \(p\) odd, for example, is equivalent to: (a) \(|G'|=p^3\) and \(|G:Z(G)|\geq p^4\), or (b) \(|G:Z(G)|=p^4\) and \(|G'|\geq p^4\) or (c) \(|G'|=p^4\) and there exists a normal subgroup \(D\) of \(G\) with \(|D|=p\) and \(|G/D:Z(G/D)|=p^3\).
scientific article; zbMATH DE number 1270873