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

Shiv Narain

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, xG=(x−1)G, but not conversely. Thus, the class of those groups satisfying Syskin’s condition forms a proper subclass of groups satisfying xG=(x−1)G. In this note, it is proved that if a group G meets the condition xG=(x−1)G, then G cannot be of odd order. As the main result, it is shown that if xG=(x−1)G holds for a centreless and non-solvable group G of order 120 such that G≠G’, then G≌S5.


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Published: 2022-09-08

How to Cite this Article:

Shiv Narain, On a non-solvable group satisfying xG=(x−1)G, Algebra Lett., 2022 (2022), Article ID 1

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.

Algebra Letters

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