Applications of the theory of Camina groups
Let \(C_1=\{1\}\), \(C_2,\dots,C_m,C_{m+1},\dots,C_n\) be all conjugacy classes of a finite group \(G\), where \(2\leq m\leq n-1\). The following hypotheses are considered. Hypothesis (H1). \(G\) has a character \(\chi\) such that (a) \(\chi(C_2)=\cdots=\chi(C_m)\), (b) for every \(\varphi\in\text{Irr}(\chi)\), \(\varphi(C_{m+1})=\cdots=\varphi(C_n)=0\). Hypothesis (H2). \(C_iC_j=C_j\) for \(i=1,\dots,m\) and \(j=m+1,\dots,n\). Theorem 1.1. Hypotheses (H1) and (H2) are equivalent. Let \(G\) satisfy (H1) and (H2). Then \(N=\bigcup^m_{i=1}C_i\) is a normal subgroup of \(G\), the kernel of \(G\). Such groups \(G\) coincide with groups that were studied by \textit{A. R. Camina} [see Isr. J. Math. 31, No. 2, 153-160 (1978; Zbl 0654.20019)]. It follows from Proposition 25.9 and the Remark 1 following it [in \textit{Ya. Berkovich} and \textit{E. Zhmud}, Characters of Finite Groups, Part 2, Transl. Math. Monogr. 181, Am. Math. Soc., Providence (1999)], that the subgroup \(N\) is solvable. In \S 3, a subclass of the class of \(V^3\)-groups, i.e., non-Abelian groups all of whose nonlinear irreducible characters assume exactly three values [see \textit{Ya. Berkovich}, \textit{D. Chillag} and \textit{E. Zhmud}, Houston J. Math. 21, No. 1, 17-28 (1995; Zbl 0833.20008)], is studied. A group \(G\) is said to be a restricted \(V^3\)-group if it is a \(V^3\)-group such that \(\ker(\chi)<G'\) for every nonlinear \(\chi\in\text{Irr}(G)\). In \S 4 the main theorem in [\textit{Ya. Berkovich}, Publ. Math. 44, No. 3-4, 225-234 (1994; Zbl 0840.20004)] is generalized slightly.
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