Arithmetical conditions on the conjugacy vector of a finite group
Copyright policy )Arithmetical conditions on the conjugacy vector of a finite group
This paper contains new properties of the numbers \(r_ G(xN)=| \{Cl_ G(g):\) \(Cl_ G(g)\cap xN\neq \emptyset \}|\) where G is a finite group and N is a normal subgroup of G. In particular, the author proves the following Theorem 1. Let \(\bar G=G/N\). Then for each \(g\in G\) one has \[ r_ G(gN)=1/| G| \cdot \sum_{x\in G}| Cl_{\bar G}(\bar g)\cap \overline{C_ G(x)}| \cdot | C_ N(x)|. \] Such arithmetical conditions are useful in the analysis of the finite groups with fixed number of conjugacy classes. [See the papers of the author and \textit{J. Vera López}, Isr. J. Math. 51, 305-338 (1985; Zbl 0582.20014); 56, 188-221 (1986; reviewed below)].
normal subgroup, finite group, number of conjugacy classes, Arithmetic and combinatorial problems involving abstract finite groups
normal subgroup, finite group, number of conjugacy classes, Arithmetic and combinatorial problems involving abstract finite groups
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