Character degrees of \(p\)-groups and pro-\(p\) groups.
In the 1970s, Isaacs conjectured that there should be a logarithmic bound for the length of solvability \(\text{dl}(G)\) of a finite \(p\)-group \(G\) with respect to the number \(\text{cd}(G)\) of different character degrees of \(G\). So far, there are just a few partial results for this conjecture. The author attacks this problem via pro-\(p\) groups. He says that a pro \(p\)-group \(G\) has property (I) if its finite quotients satisfy Isaacs' conjecture, namely if there exists a real number \(D(G)\) that depends only on \(G\) such that for any open normal subgroup \(N\) of \(G\), \(\text{dl}(G/N)\leq\log_2|\text{cd}(G/N)|+D(G)\). He proves that any \(p\)-adic analytic pro-\(p\) group has property (I). -- The proof uses Howe's Kirillov theory for compact \(p\)-adic analytic groups to deal with the case when \(G\) is a uniform group and \(N=G_n\) is a dimensional subgroup. A second result concerns the first congruence subgroup \(G\) of a classical Chevalley group \(\mathbb{G}\) with respect to the local ring \(\mathbb{F}_p[\![t]\!]\); the author shows that if \(\text{Lie}(\mathbb{G})(\mathbb{F}_p)\) has a non-degenerate Killing form, then \(G=\ker(\mathbb{G}(\mathbb{F}_p[\![t]\!])\to\mathbb{G}(\mathbb{F}_p))\) has property (I).
scientific article; zbMATH DE number 2164792