On non-solvable Camina pairs.
The authors consider the structure of a class of finite groups first introduced by the reviewer in 1978 [Isr. J. Math. 31, No. 2, 153-160 (1978; Zbl 0654.20019)]. A group \(G\) is said to have a Camina kernel if it has a nontrivial proper normal subgroup \(N\) such that for all \(g\in G\setminus N\) and \(k\in N\), \(gk\) is conjugate to \(g\). Such a pair \((G,N)\) is called a Camina pair. The reviewer showed that one of three possibilities occurs: (i) \(G\) is a Frobenius group with kernel \(N\); (ii) \(N\) is a \(p\)-group; or (iii) \(G/N\) is a \(p\)-group. \textit{D. Chillag, A. Mann} and \textit{C. M. Scoppola} proved that if \(G/N\) is a \(p\)-group then \(G\) is soluble [Isr. J. Math. 62, No. 3, 269-282 (1988; Zbl 0654.20020)]. In this paper the authors consider the situation when \(G\) is not soluble. They prove amongst other results, the following theorem: Let \((G,N)\) be a Camina pair and assume that \(G\) is not soluble then \(N\) is a \(p\)-group and one of the following holds (a) \((G/\mathrm{O}_p(G))^\infty\cong\mathrm{SL}(2,p^e)\) with \(p^e>3\); (b) \((G/\mathrm{O}_p(G))^\infty\cong\mathrm{SL}(2,5)\) with \(p=3\); (c) \((G/\mathrm{O}_p(G))^\infty\cong\mathrm{SL}(2,13)\) with \(p=3\); (d) \((G/\mathrm{O}_p(G))^\infty\cong\mathrm{SL}(2,5)\) with \(p\geq 7\) and \((\mathrm{S}(G),N)\) is a Camina pair; \(G/\mathrm{S}(G)\cong A_5\) or \(S_5\). In the statement of the theorem \(\mathrm{S}(G)\) maximal soluble normal subgroup of \(G\) and \(G^\infty\) is the last term of the derived series of \(G\). -- They also give an example of a group \(G\) with a Camina pair where \(\mathrm{O}_p(G)\) has order \(5^5\) and \(G/\mathrm{O}_p(G)\cong\mathrm{SL}(2,5)\). The proof depends on the results of \textit{P. Fleischmann, W. Lempken} and \textit{P. H. Tiep} [J. Algebra 188, No. 2, 547-579 (1997; Zbl 0896.20007)] which describe the structure of a group, \(G\), which acts on a module over a finite field of characteristic \(p\) so that each element of order prime to \(p\) in \(G\) acts without non-trivial fixed points. It is known that if \((G,N)\) is a Camina pair and \(N\) is a \(p\)-group then the action of \(G/\Phi(N)\) has this property. This paper uses these results to show that the possible structure of a non-soluble group with a Camina pair is quite restricted. They also show how a group with two irreducible characters whose product is a positive linear combination of the two is related to groups with Camina pairs and deduce a nice theorem about such groups.
scientific article; zbMATH DE number 5655008
scientific article