| Title:
|
Dihedral-like constructions of automorphic loops (English) |
| Author:
|
Aboras, Mouna |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
269-284 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Automorphic loops are loops in which all inner mappings are automorphisms. We study a generalization of the dihedral construction for groups. Namely, if $(G,+)$ is an abelian group, $m\geq 1$ and $\alpha \in \operatorname{Aut}(G)$, let $\operatorname{Dih} (m,G,\alpha )$ be defined on $\mathbb Z_m\times G$ by \begin{equation*} (i,u)(j,v) = (i\oplus j,\,((-1)^{j}u + v)\alpha^{ij}). \end{equation*} The resulting loop is automorphic if and only if $m=2$ or ($\alpha^2=1$ and $m$ is even). The case $m=2$ was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský. We present several structural results about the automorphic dihedral loops in both cases. (English) |
| Keyword:
|
dihedral automorphic loop |
| Keyword:
|
automorphic loop |
| Keyword:
|
inner mapping group |
| Keyword:
|
multiplication group |
| Keyword:
|
nucleus |
| Keyword:
|
commutant |
| Keyword:
|
center |
| Keyword:
|
commutator |
| Keyword:
|
associator subloop |
| Keyword:
|
derived subloop |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 06391543 |
| idMR:
|
MR3225610 |
| . |
| Date available:
|
2015-01-19T10:47:30Z |
| Last updated:
|
2016-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143807 |
| . |
| Reference:
|
[1] Kinyon M.K., Kunen K., Phillips J.D., Vojtěchovský P.: The structure of automorphic loops.to appear in Transactions of the American Mathematical Society. MR 2302693 |
| Reference:
|
[2] Bruck R.H.: A Survey of Binary Systems.Springer, 1971. Zbl 0141.01401, MR 0093552 |
| Reference:
|
[3] Bruck R.H., Paige L.J.: Loops whose inner mappings are automorphisms.Ann. of Math. 2 63 (1956), 308–323. Zbl 0074.01701, MR 0076779, 10.2307/1969612 |
| Reference:
|
[4] Johnson K.W., Kinyon M.K., Nagy G.P., Vojtěchovský P.: Searching for small simple automorphic loops.LMS J. Comut. Math. 14 (2011), 200–213. Zbl 1225.20052, MR 2831230, 10.1112/S1461157010000173 |
| Reference:
|
[5] Jedlička P., Kinyon M.K., Vojtěchovský P.: The structure of commutative automorphic loops.Trans. Amer. Math. Soc. 363 (2011), no. 1, 365–384. Zbl 1215.20060, MR 2719686, 10.1090/S0002-9947-2010-05088-3 |
| Reference:
|
[6] Jedlička P., Kinyon M.K., Vojtěchovský P.: Constructions of commutative automorphic loops.Comm. Algebra 38 (2010), no. 9, 3243–3267. Zbl 1209.20069, MR 2724218, 10.1080/00927870903200877 |
| Reference:
|
[7] Aboras M.: Dihedral-like constructions of automorphic loops.Thesis, in preparation. |
| . |
