Abstract
Let G be a finite p-group. Assume that \(\nu (G)\) and \(\nu _c(G)\) denote the number of conjugacy classes of non-normal subgroups and non-normal cyclic subgroups of G, respectively. In this paper, we completely classify the finite p-groups with \(\nu _c=p\) or \(p+1\) for an odd prime number p. Also, we classify the groups G with \(\nu (G)=\nu _c(G)=p^i, i\ge 1\).
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References
Berkovich, Y.: Groups of Prime Power Order I. Springer, Berlin (2008)
Blackburn, N.: Finite groups in which the normal subgroups have non-trivial intersection. J. Algebra 3, 30–37 (1966)
Brandl, R.: Groups with few non-normal subgroups. Commun. Algebra 6, 2091–2098 (1995)
Brandl, R.: Conjugacy classes of non-normal subgroups of finite p-groups. Isr. J. Math. 195, 473–479 (2013)
Bozikov, Z., Janko, Z.: A complete classification of finite p-groups all of whose non-cyclic subgroups are normal. Glasnik Mat. 44(1), 177–185 (2009)
Cohn, J.H.E.: On n-sum groups. Math. Scand. 75, 44–58 (1994)
Fernandez-Alcober, G.A., Legarreta, L.: The finite \(p\)-groups with \(p\) conjugacy classes of non-normal subgroups. Isr. J. Math. 180, 189–192 (2010)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Li, L., Qu, H.: The number of conjugacy classes of non-normal subgroups of finite p-groups. J. Algebra 1 (2016)
Mousavi, H.: On finite groups with few non-normal subgroups. Commun. Algebra 7, 3143–3151 (1999)
Mousavi, H.: Nilpotent groups with three conjugacy classes of non-normal subgroups. Bull. Iran. Math. Soc. 5, 1291–1300 (2014)
Mousavi, H.: Non-nilpotent groups with three conjugacy classes of non-normal subgroups. Int. J. Group Theory 3(2), 1–7 (2014)
Mousavi, H., Ahmadi, H.: A bound on the number of conjugacy classes of non-normal cyclic subgroups of a finite \(p\)-group (submitted)
Oggionni, D., Ponzoni, G., Zambelli, V.: Groups with few non-normal cyclic subgroups. Note Mat. 30(2), 121–133 (2010)
Passman, D.S.: Non-normal subgroups of \(p\)-groups. J. Algebra 15(3), 352–370 (1970)
Poland, J., Rhemtulla, A.: The number of conjugacy classes of non-normal subgroups in nilpotent groups. Commun. Algebra 24(10), 3237–3245 (1996)
Redei, L.: Das Schiefe Produkt in der Gruppentheorie. Comment. Math. Helv. 20, 225–264 (1947)
Shirong, L.: The number of conjugacy classes of non-normal cyclic subgroups in nilpotent groups of odd order. J. Group Theory 1, 165–171 (1998)
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Communicated by Mohammad Zarrin.
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Mousavi, H. On the Conjugacy Classes of Cyclic Non-normal Subgroups. Bull. Iran. Math. Soc. 48, 73–92 (2022). https://doi.org/10.1007/s41980-020-00502-6
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DOI: https://doi.org/10.1007/s41980-020-00502-6