Let \(G\) be a finite \(p\)-group, \(p\) a prime, \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}\). According to Isaacs’ result, every finite set \(\mathcal S\) of powers of \(p\) containing \(1\), equals \(\text{cd}(G)\) for some \(p\)-group \(G\) of class \(\leq 2\) (in what follows we assume that \(1\in\mathcal S\) always). The set \(\mathcal S\) is class bounding if there exists \(n\in\mathbb{N}\) such that every \(p\)-group \(G\) with \(\text{cd}(G)=\mathcal S\) has class at most \(n\). The set \(\mathcal S\) is ‘strongly class bounding’ if every subset of \(\mathcal S\) that contains \(1\) is class bounding. The authors ask if any strongly class bounding set is also class bounding. There are a lot of papers devoted to criteria for \(\mathcal S\) to be class bounding. This paper is an interesting contribution to this theme. In their investigation the authors use the theory of pro-\(p\)-groups. Below we present some typical results.
Let \(\epsilon_p=0\) if \(p>2\) and \(\epsilon_2=1\).
Theorem B. Let \(p^j=\min\{d\mid d\in{\mathcal S}-\{1\}\}p\). If \(|{\mathcal S}|\leq j+2-\epsilon_p\), then \(\mathcal S\) is class bounding.
Clearly, \(\mathcal S\) of Theorem B is strongly class bounding.
Corollary C. Given any \(n\in\mathbb{N}\) and a prime \(p\), there exists a \(p\)-group \(G\) such that \(\text{cd}(G)\) is class bounding and the derived length of \(G\) equals \(n\).
Theorem E. If \(\max\{d\mid d\in{\mathcal S}\}<p^j\) for some \(j>1\), then the set \({\mathcal S}'={\mathcal S}\cup\{p^j,\dots,p^{2j+1}\}\) is not class bounding.
It follows from Theorem E that the set \({\mathcal S}=\{1,p^2,p^3,p^4,\dots,p^7\}\) is not class bounding and \(p\not\in\mathcal S\), answering a question of Isaacs and Slattery. In Theorem A non strongly class bounding sets are characterized.
In conclusion \(6\) open questions are formulated. We present only two of them. Is \(\{1,2^2,2^4,2^5\}\) class bounding? Is it true that if \({\mathcal S}\supseteq\{1,p^j,\dots,p^{2j+1}\}\) for some \(j>1\), then \(\mathcal S\) is not class bounding?