Classification of finite groups according to the number of conjugacy classes. II
This paper continues the authors' work on the classification of finite groups with a fixed number of conjugacy classes. Let G be a finite group. Let r(G) denote the number of conjugacy classes of G, \(\beta\) (G) the number of minimal normal subgroups of G, and \(\alpha\) (G) the number of conjugacy classes of G not contained in the socle S(G). The main result of the paper is the classification of the family \(\phi_{11}=\{G|\beta (G)=r(G)-11\}\). In part I of the paper this was done for \(\phi_ j\), \(1\leq j\leq 10\) [ibid. 51, 305-338 (1985; Zbl 0582.20014)]. Based upon the description of \(\phi_{11}\) the authors classify all finite groups satisfying one of the following conditions: 1) \(r(G)=12.\) 2) 13\(\leq r(G)\leq 20\), \(\beta\) (G)\(\geq r(G)-11.\) 3) r(G)\(\geq 21\), r(G)-11\(\leq \beta (G)\leq r(G)-1.\) 4) 13\(\leq r(G)\leq 20\), \(\beta\) (G)\(\leq r(G)-12\), G is no nonabelian simple group and either \(0\leq \alpha (G)\leq 4\) or S(G) is solvable and \(5\leq \alpha (G)\leq 10.\) Reviewer's remarks: 1. The classification of finite groups with \(r(G)=12\) was obtained independently by \textit{Ya. G. Berkovich} [Publ. Math. 33, 107-123 (1986; reviewed below)]. 2. There are some inaccuracies in the paper reviewed. In the classification of finite groups G with 13\(\leq r(G)\leq 20\), \(\beta\) (G)\(\leq r(G)-12\) and \(0\leq \alpha (G)\leq 4\) the condition ``G is no nonabelian simple group'' is missing. In Table 1 the group \(Z_ 2\times (E_ 9\leftthreetimes_ fZ_ 2)\) is missing. There are two nonisomorphic groups of type \(Z_ 3\leftthreetimes SD_{16}\) with \(C(Z_ 3)\cong Z_ 3\times D_ 8\) and \(Z_ 3\times Q_ 8\), respectively. However, this fact is not mentioned in Table 1.
scientific article; zbMATH DE number 4033989
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