Average order in regular wreath products

Sarkar, Supravat

doi:10.22108/ijgt.2023.135927.1818

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	Average order in regular wreath products
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 25 شهریور 1402 اصل مقاله (412.88 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2023.135927.1818
نویسنده
Supravat Sarkar^*
Mathematics Ph.D. student in Princeton University, USA
چکیده
We obtain an exact formula for the average order of elements of regular wreath product of two finite groups. Then focussing our attention on $p$-groups for primes $p$, we give an estimate for the verage order of a wreath product $A\wr B$ in terms of maximum order of elements of $A$ and average order of $B$ and an exact formula for the distribution of orders of elements of $A\wr B.$ Finally, we show how wreath products can be used to find several rational numbers which are limits of average orders of a sequence of $p$-groups with cardinalities going to infinity.
کلیدواژه‌ها
average order؛ wreath product؛ maximum order؛ order distribution؛ semidirect product

مراجع
[1] H. Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl., 10 (2011) 187–190. [2] H. Amiri, S. M. Jafarian Amiri and I. M. Isaacs, Sums of element orders in finite groups, Comm. Algebra, 37 (2009) 2978–2980. [3] J. D. P. Meldrum, Wreath products of groups and semigroups, Pitman Monographs and Surveys in Pure and pplied Mathematics, 74, Longman, Harlow, 1995. [4] M. Herzog, P. Longobardi and M. Maj, New criteria for solvability, nilpotency and other properties of finite groups in terms of the order elements or subgroups, Int. J. Group Theory, 12 (2023) 35–44. [5] M.l Herzog, Pa. Longobardi mand M. Maj, Two new criteria for solvability of finite groups, J. Algebra, 511 (2018) 215–226. [6] M. S. Lazorec and M. Tarnauceanu , On the average order of a finite group, J. Pure Appl. Algebra, 227 (2023) 9 pp.
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Average order in regular wreath products