A Class of Loops with the Isotopy-Isomorphy Property
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 589-592
Voir la notice de l'article provenant de la source Cambridge University Press
A loop L has the isotopy-isomorphy property provided each loop isotopic to L is isomorphic to L. A familiar problem is that of characterizing those loops having this property.It is well known (1, p. 56) that the loop isotopes of (L, ·) are those loops L(a, b, *) defined by x * y = x/b·a\y for some a, b in L. In this paper we first show (Corollary to Theorem 1) that a loop L with identity element 1 has the isotopy-isomorphy property if L is isomorphic to 1,(1, x) and to L(x, 1) for each x in L. We then determine necessary and sufficient conditions (Theorems 2 and 3) for L to be isomorphic to these isotopes under translations (i.e. permutations of the form xv = cx or xv = xc for c fixed).
Wilson, Eric L. A Class of Loops with the Isotopy-Isomorphy Property. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 589-592. doi: 10.4153/CJM-1966-057-0
@article{10_4153_CJM_1966_057_0,
author = {Wilson, Eric L.},
title = {A {Class} of {Loops} with the {Isotopy-Isomorphy} {Property}},
journal = {Canadian journal of mathematics},
pages = {589--592},
year = {1966},
volume = {18},
number = {1},
doi = {10.4153/CJM-1966-057-0},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-057-0/}
}
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