Every finite nilpotent loop has a supernilpotent loop as reduct
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918226315706368 |
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| author | Kompatscher, Michael Mayr, Peter |
| author_facet | Kompatscher, Michael Mayr, Peter |
| contents | A basic fact taught in undergraduate algebra courses is that every finite nilpotent group is a direct product of $p$-groups. Already Bruck observed that this does not generalize to loops. In particular, there exist nilpotent loops of size $6$ which are not direct products of loops of size $2$ and $3$. Still we show that every finite nilpotent loop $(A,\cdot)$ has a binary term operation $*$ such that $(A,*)$ is a direct product of nilpotent loops of prime power order, i.e., $(A,*)$ is supernilpotent. As an application we obtain that every nilpotent loop of order $pq$ for primes $p,q$ has a finite basis for its equational theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12484 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Every finite nilpotent loop has a supernilpotent loop as reduct Kompatscher, Michael Mayr, Peter Group Theory 20N05 A basic fact taught in undergraduate algebra courses is that every finite nilpotent group is a direct product of $p$-groups. Already Bruck observed that this does not generalize to loops. In particular, there exist nilpotent loops of size $6$ which are not direct products of loops of size $2$ and $3$. Still we show that every finite nilpotent loop $(A,\cdot)$ has a binary term operation $*$ such that $(A,*)$ is a direct product of nilpotent loops of prime power order, i.e., $(A,*)$ is supernilpotent. As an application we obtain that every nilpotent loop of order $pq$ for primes $p,q$ has a finite basis for its equational theory. |
| title | Every finite nilpotent loop has a supernilpotent loop as reduct |
| topic | Group Theory 20N05 |
| url | https://arxiv.org/abs/2409.12484 |