The conjugacy vector of a \(p\)-group of maximal class (Q1327508)

A finite \(p\)-group \(G\) is of maximal class if \(| G|=p^ m\), \(m \geq 2\) and \(\text{class}(G)=m-1\). All the groups considered in this paper are \(p\)-groups of maximal class. If \(a \geq 1\), the family of groups whose largest abelian normal subgroup is \(G_ a=[G,\dots, G]\) is denoted by \({\mathcal G}_ a\). The number of conjugacy classes of \(G\) is denoted by \(r(G)\) and \(c=c(G)\) is the degree of commutativity of \(G\), i.e. \(c(G)=\max \{k \leq m-2\mid [G_ i, G_ j] \leq G_{i+j+k},\;\forall i,j \geq 1\}\). In the main result of this paper it is proved that if \(| G|=p^ m\) and \(G \in {\mathcal G}_ a\), then \[ p^{m- 2a}+(p^ 2-1)\;((a-1) p^{c-1}+1) \leq r(G) \leq p^{m-3}+p^ c- p^{c-1}+p^ 2-1, \] and the authors characterize the groups for which equality holds.