Loops with squares in two nuclei
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918172189261824 |
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| author | Kinyon, Michael Phillips, J. D. |
| author_facet | Kinyon, Michael Phillips, J. D. |
| contents | Although little can be gleaned about a loop with the property that its squares are, say, left nuclear ($xx\cdot yz = (xx\cdot y)z$), if its squares are also, say, middle nuclear ($(x\cdot yy)z = x(yy\cdot z)$), then the loop exhibits more structure than one might initially guess. Loops with squares in (at least) two nuclei include many well known classes of loops, such as C loops and extra loops, and not so well known classes such left C loops. In any loop with, say, left and middle nuclear squares, the intersection of the left and middle nuclei is a normal subloop; hence such a loop is simple if and only if it is a group or a simple unipotent loop. Loops in which squaring is a centralizing endomorphism have even more structure; they are power-associative, and a torsion loop in that class is a direct product of a loop of $2$-elements and a loop of elements of odd order. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_19961 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Loops with squares in two nuclei Kinyon, Michael Phillips, J. D. Group Theory 20N05 Although little can be gleaned about a loop with the property that its squares are, say, left nuclear ($xx\cdot yz = (xx\cdot y)z$), if its squares are also, say, middle nuclear ($(x\cdot yy)z = x(yy\cdot z)$), then the loop exhibits more structure than one might initially guess. Loops with squares in (at least) two nuclei include many well known classes of loops, such as C loops and extra loops, and not so well known classes such left C loops. In any loop with, say, left and middle nuclear squares, the intersection of the left and middle nuclei is a normal subloop; hence such a loop is simple if and only if it is a group or a simple unipotent loop. Loops in which squaring is a centralizing endomorphism have even more structure; they are power-associative, and a torsion loop in that class is a direct product of a loop of $2$-elements and a loop of elements of odd order. |
| title | Loops with squares in two nuclei |
| topic | Group Theory 20N05 |
| url | https://arxiv.org/abs/2510.19961 |