Triangles in the graph of conjugacy classes of normal subgroups
Abstract
[EN] Let G be a finite group and N a normal subgroup of G. We determine the structure of N when the graph G(N), which is the graph associated to the conjugacy classes of G contained in N, has no triangles and when the graph consists in exactly one triangle. ; The research of the first and second authors is supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, Grant P11B2015-77. ; Beltrán, A.; Felipe Román, MJ.; Melchor, C. (2017). Triangles in the graph of conjugacy classes of normal subgroups. Monatshefte für Mathematik. 182(1):5-21. https://doi.org/10.1007/S00605-015-0866-9 ; S ; 5 ; 21 ; 182 ; 1 ; Bertram, E.A., Herzog, M., Mann, A.: On a graph related to conjugacy classes of groups. Bull. London Math. Soc. 22(6), 569-575 (1990) ; Beltrán, A., Felipe, M.J., Melchor, C.: Graphs associated to conjugacy classes of normal subgroups in finite groups. J. Algebra 443, 335-348 (2015) ; Camina, A.R.: Arithmetical conditions on the conjugacy class numbers of a finite group. J. London Math. Soc. 2(5), 127-132 (1972) ; Deaconescu, M.: Classification of finite groups with all elements of prime order. Proc. Am. Math. Soc. 106(3), 625-629 (1989) ; Doerk, K., Hawkes, T.: Finite soluble groups. de Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter, Berlin (1992) ; Fang, M., Zhang, P.: Finite groups with graphs containing no triangles. J. Algebra 264(2), 613-619 (2003) ; Higman, G.: Finite groups in which every element has prime power order. J. London Math. Soc. 32, 335-342 (1957) ; Manz, O., Wolf, T.R.: Representations of solvable groups. Cambridge Univ. Press, Cambridge (1993) ; Riese, U., Shahabi, M.A.: Subgroups which are the union of four conjugacy classes. Commun. Algebra 29(2), 695-701 (2001) ; Shahryari, M., Shahabi, M.A.: Subgroups which are the union of three conjugate classes. J. Algebra 207(1), 326-332 (1998) ; The GAP Group.: GAP–groups, algorithms and programming, Vers. 4.4.12. (2008). ...
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