New lower bounds for the number of conjugacy classes in finite nilpotent groups

Document Type : Research Paper

Author

‎Department of‎ ‎Mathematics‎, ‎University of Hawaii‎, ‎Honolulu‎, ‎HI 96822‎, ‎USA

Abstract

P‎. ‎Hall's classical equality for the number of conjugacy classes in $p$-groups yields $k(G) \ge (3/2) \log_2 |G|$ when $G$ is nilpotent‎. ‎Using only Hall's theorem‎, ‎this is the best one can do when $|G| = 2^n$‎. ‎Using a result of G.J‎. ‎Sherman‎, ‎we improve the constant $3/2$ to $5/3$‎, ‎which is best possible across all nilpotent groups and to $15/8$ when $G$ is nilpotent and $|G| \ne 8,16$‎. ‎These results are then used to prove that $k(G) > \log_3(|G|)$ when $G/N$ is nilpotent‎, ‎under natural conditions on $N \trianglelefteq G$‎. ‎Also‎, ‎when $G'$ is nilpotent of class $c$‎, ‎we prove that $k(G) \ge (\log |G|)^t$ when $|G|$ is large enough‎, ‎depending only on $(c,t)$‎.

Keywords

Main Subjects

References

[1] B. Baumeister, A. Maróti and H. P. Tong-Viet, Finite groups have more conjugacy classes, Forum Math., 29 (2017)
259–275.
[2] E. A. Bertram, Large centralizers in finite solvable groups, Israel J. Math., 47 (1984) 335–344.
[3] E. A. Bertram, Lower bounds for the number of conjugacy classes in finite solvable groups, Israel J. Math., 75
(1991) 243–255.
[4] E. A. Bertram, Lower bounds for the Number of conjugacy classes in finite groups, Ischia Group Theory 2004,
95–117, Contemp. Math., 402, Israel Math. Conf. Proc., Amer. Math. Soc., Providence, 2006.
[5] E. A. Bertram, New reductions and logorithmic lower bounds for the number of conjugacy classes in finite groups,
Bull. Aust. Math. Soc., 87 (2013) 406–424.
[6] Hans Ulrich Besche, Bettina Eick and E. A. O’Brien, A millenium project: constructing small groups, Inter. J.
Algebra Comput., 12 (2002) 623–644.
[7] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24
(1997) 235–265.
[8] N. Boston and J. L.Walker, 2-Groups with few conjugacy classes, Proc. Edinburgh Math. Soc., 43 (2000) 211–217.
[9] M. Cartwright, The number of conjugacy classes of certain finite groups, J. Lond. Math Soc.(2), 36 (1985) 393–404.
[10] A. Jaikin-Zapirain, On the number of conjugacy classes of finite nilpotent groups, Adv. Math., 227 (2011) 1129–1143.
[11] B. Huppert, Endliche gruppen I, (Springer, Berlin, 1967).
[12] A. Mann, Conjugacy classes in finite groups, Israel J. Math., 31 (1978) 78–84.
[13] E. A. O’Brien, The p-group generation algorithm, J. Symbolic Comput., 9 (1990) 677–698.
[14] G. J. Sherman, A lower bound for the number of conjugacy classes in a finite nilpotent group, Pacific J. Math, 80
(1979) 253–254.
[15] A. Vera-Lopez and J. Sangroniz, The finite groups with thirteen and fourteen conjugacy classes, Math. Nach., 280
(2007) 676–694.
[16] A. Vera-Lopez and J. Vera-Lopez, Classification of finite groups according to the number of conjugacy classes, Israel
J. Math., 51 (1985) 305–338.
[17] A. Vera-Lopez and J. Vera-Lopez, Classification of finite groups according to the number of conjugacy classes II,
Israel J. Math., 56 (1986) 188–221.

Files

Share

How to cite

Statistics