On a non-solvable group satisfying xG=(x−1)G

Narain, Shiv

doi:10.28919/al/7634

Algebra Lett. 2022 (2022), Article ID 1

On a non-solvable group satisfying xG=(x−1)G

Shiv Narain

PDF (99.4 KB) https://doi.org/10.28919/al/7634

Abstract

A group G satisfies Syskin’s condition if elements of same order are conjugates. If a group G satisfies Syskin’s condition, then each element and its inverse are conjugate to each other, i.e., for all x∈G, x^G=(x⁻¹)^G, but not conversely. Thus, the class of those groups satisfying Syskin’s condition forms a proper subclass of groups satisfying x^G=(x⁻¹)^G. In this note, it is proved that if a group G meets the condition x^G=(x⁻¹)^G, then G cannot be of odd order. As the main result, it is shown that if x^G=(x⁻¹)^G holds for a centreless and non-solvable group G of order 120 such that G≠G’, then G≌S5.

Published: 8 September 2022
Volume: 2022 (2022)
License: CC BY 3.0

Copyright © 2022 Shiv Narain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Cite this article

S. Narain, On a non-solvable group satisfying xG=(x−1)G, Algebra Lett. 2022 (2022), 1. https://doi.org/10.28919/al/7634

BibTeX

@article{Narain2022,
  author  = {Narain, Shiv},
  title   = {On a non-solvable group satisfying xG=(x−1)G},
  journal = {Algebra Lett.},
  volume  = {2022},
  year    = {2022},
  pages   = {1},
  doi     = {10.28919/al/7634}
}