Structure of finite groups with restrictions on the set of conjugacy classes sizes (Q6591975)

Let \(G\) be a finite group with no central direct factors and let \(N(G)=\big \{ |g^{G}| \; \big | \; g \in G \big \}\) be the set of conjugacy classes sizes of \(G\). If \(G=A \times B\), then it is easy to check that \(N(G)=N(A)\times N(B)\). In the paper under review the author is interested in determining when the inverse implication may be valid.\N\NThe main result of this paper is Theorem 0.2: Let \(\Omega\) be a set of integers and \(\Gamma(\Omega\setminus \{1\})\) be disconnected, and \(n\) be a positive integer such that \(\mathrm{gcd}(n, \alpha)=1\) for each \( \alpha \in \Omega \setminus \{1\}\). Let \(G\) be a finite group such that \(N(G)=\Omega \times \{1, n\}\). Then \(G=A\times B\), where \(N(A)=\Omega\), \(N(B)=\{1,n \}\) and \(n\) is a prime power.