Abstract
We prove that if G is a finite group, N is a normal subgroup, and there is a prime p so that all the elements in G ∖ N have p-power order, then either G is a p-group or G = PN where P is a Sylow p-subgroup and (G,P,P ∩ N) is a Frobenius–Wielandt triple. We also prove that if all the elements of G ∖ N have prime power orders and the orders are divisible by two primes p and q, then G is a {p,q}-group and G/N is either a Frobenius group or a 2-Frobenius group. If all the elements of G ∖ N have prime power orders and the orders are divisible by at least three primes, then all elements of G have prime power order and G/N is nonsolvable.
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References
Bianchi, M., Camina, R.D., Lewis, M.L., Pacifici, E.: Conjugacy classes of maximal cyclic subgroups. Submitted for publication. arXiv:2201.05637 (2022)
Brandl, R.: Finite groups all of whose elements are of prime power order. Boll. Un. Mat. Ital. A (5) 18, 491–493 (1981)
Bubboloni, D.: Coverings of the symmetric and alternating groups. arXiv:1009.3866 (2010)
Burkett, S., Lewis, M.L.: A Frobenius group analog for Camina triples. J. Algebra 568, 160–180 (2021)
Cheng, K.N., Deaconescu, M., Lang, M.-L., Shi, W.J.: Corrigendum and addendum to: “Classification of finite groups with all elements of prime order” [Proc. Amer. Math. Soc. 106 (1989), no. 3, 625–629; MR0969518 (89k:20038)] by Deaconescu. Proc. Amer. Math. Soc. 117, 1205–1207 (1993)
Deaconescu, M.: Classification of finite groups with all elements of prime order. Proc. Amer. Math. Soc. 106, 625–629 (1989)
Espuelas, A.: The complement of a Frobenius–Wielandt group. Proc. Lond. Math. Soc. (3) 48, 564–576 (1984)
Havas, G., Vaughan-Lee, M.: On counterexamples to the Hughes conjecture. J. Algebra 322, 791–801 (2009)
Heineken, H.: On groups all of whose elements have prime power order. Math. Proc. R. Ir. Acad. 106A, 191–198 (2006)
Higman, G.: Finite groups in which every element has prime power order. J. Lond. Math. Soc. (2) 32, 335–342 (1957)
Huppert, B.: Endliche Gruppen, vol. I. Springer, Berlin (1967)
Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1976)
Isaacs, I.M.: Finite Group Theory. Graduate Studies in Mathematics, vol. 92. American Mathematical Society, Providence, RI (2008)
Knapp, W., Schmid, P.: A note on Frobenius groups. J. Group Theory 12, 393–400 (2009)
Qian, G.H.: Finite groups with many elements of prime order (in Chinese). J. Math. (Wuhan) 25, 115–118 (2005)
Scoppola, C.M.: Groups of prime power order as Frobenius-Wielandt complements. Trans. Amer. Math. Soc. 325, 855–874 (1991)
Shi, W., Yang, W.: The finite groups of all whose elements are of prime power order (in Chinese). J. Yunnan Educational Coll. Ser. B 1, 2–10 (1986). English translation at arXiv:2003.09445. 2020
Suzuki, M.: On a class of doubly transitive groups. Ann. Math. (2) 75, 105–145 (1962)
Wielandt, H.: ÜBer die Existenz von Normalteilern in endlichen Gruppen. Math. Nachr. 18, 274–280 (1958)
Williams, J.S.: Prime graph components of finite groups. J. Algebra 69, 487–513 (1981)
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Lewis, M.L. Groups Having all Elements off a Normal Subgroup with Prime Power Order. Vietnam J. Math. 51, 577–587 (2023). https://doi.org/10.1007/s10013-022-00591-2
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DOI: https://doi.org/10.1007/s10013-022-00591-2