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Groups,
conjugation and powers
Markus Szymik and Torstein Vik |
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Vol. 18 (2025), No. 3, 387–399
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Abstract |
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We introduce the notion of the power quandle of a group, an algebraic structure that forgets the multiplication but keeps the conjugation and the power maps. Compared with plain quandles, power quandles are much better invariants of groups. We show that they determine the central quotient of any group and the center of any finite group. Any group can be canonically approximated by the associated group of its power quandle, which we show to be a central extension, with a universal property and a computable kernel. This allows us to present any group as a quotient of a group with a power-conjugation presentation by an abelian subgroup that is determined by the power quandle and low-dimensional homological invariants. |
Keywords
groups, quandles, power operations
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Mathematical Subject Classification
Primary: 20A05, 20F05, 20J06, 20N02
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Milestones
Received: 29 April 2022
Accepted: 28 February 2024
Published: 28 April 2025
Communicated by Józef H. Przytycki
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Authors |
| Markus Szymik | |
| Department of Mathematical
Sciences Norwegian University of Science and Technology Trondheim Norway |
School of Mathematical and Physical
Sciences University of Sheffield Sheffield United Kingdom |
| Torstein Vik | |
| Department of Mathematical
Sciences Norwegian University of Science and Technology Trondheim Norway |
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| © 2025 MSP (Mathematical Sciences Publishers). |
