Groups whose same-order types are arithmetic progressions

Document Type : Research Paper

Authors

Department of Mathematics, Hubei Minzu University, Enshi, Hubei, China

Abstract

For any group $G$, define $g\sim h$ if $g,h\in G$ have the same order. The set of sizes of the equivalent classes with respect to this relation is called the same-order type of $G$. In this short note we prove that there is no finite group whose same-order type is an arithmetic progression of length $4$. This answered an open problem posed by Lazorec and  Tˇarnˇauceanu

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References

[1] R. Shen, On groups with given same-order types, Comm. Algebra, 40 no. 6 (2012) 2140–2150.
[2] M.-S. Lazorec, M. Tǎrnǎuceanu, Groups whose same-order types are arithmetic progressions, Quaest. Math., 45 no. 8 (2022) 1309–1316.
[3] F. G. Frobenius, Verallgemeinerung des Sylowschen Satzes, Sitzungsber. K. Preuss. Akad. Wiss. Berlin, II (1895) 981–993.
[4] L. Weisner, On the number of elements of a group which have a power in a given conjugate set, Bull. Amer. Math. Soc., 31 no. 9 (1925) 492–496.
[5] Y. Berkovich, Z. Janko, Groups of prime power order, 2, De Gruyter Expositions in Mathematics, 47, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.
[6] The GAP Group, GAP-Groups, Algorithms and Programming, Vers.4.4.12 (2008). http://www.gap-system.org/
International Journal of Group Theory

Articles in Press, Accepted Manuscript
Available Online from 12 September 2024

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History

  • Receive Date: 21 May 2024
  • Revise Date: 05 September 2024
  • Accept Date: 12 September 2024
  • Published Online: 12 September 2024

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