Quasigroup Automorphisms and the Norton-stein Complex
نویسندگان
چکیده
Suppose that d > 1 is the largest power of two that divides the order of a finite quasigroup Q. It then follows that each automorphism of Q must contain a cycle of length not divisible by d in its disjoint cycle decomposition. The proof is obtained by considering the action induced by the automorphism on a certain orientable surface originally described in a more restricted context by Norton and Stein.
منابع مشابه
A graph-theoretic approach to quasigroup cycle numbers
Norton and Stein associated a number with each idempotent quasigroup or diagonalized Latin square of given finite order n, showing that it is congruent mod 2 to the triangular number T (n). In this paper, we use a graph-theoretic approach to extend their invariant to an arbitrary finite quasigroup. We call it the cycle number, and identify it as the number of connected components in a certain g...
متن کاملUNIVERSITATIS APULENSIS No 15 / 2008 G - N - QUASIGROUPS
In this paper we present criteria for an n-quasigroup to be isotopic to an n-group. We call a such n-quasigroup G−n-quasigroup. Applications to functional equations on quasigroups are presented in a subsequent paper. 2000 Mathematics Subject Classification: 20N15. Some important n-quasigroup classes are the following. An n-quasigroup (A,α) of the form α(x1 ) = n ∑ i=1 fi(xi)+a, where (A,+) is a...
متن کاملOn pseudoisomorphy and distributivity of quasigroups
A repeated bijection in an isotopism of quasigroups is called a companion of the third component. The last is called a pseudoisomorphism with the companion. Isotopy coincides with pseudoisomorphy∗ in the class of inverse property loops and with isomorphy in the class of commutative inverse property loops. This result is a generalization of the corresponding theorem for commutative Moufang loops...
متن کاملThe spectrum for quasigroups with cyclic automorphisms and additional symmetries
We determine necessary and sufficient conditions for the existence of a quasigroup of order n having an automorphism consisting of a single cycle of length m and n − m fixed points, and having any combination of the additional properties of being idempotent, unipotent, commutative, semi-symmetric or totally symmetric. Quasigroups with such additional properties and symmetries are equivalent to ...
متن کاملPalindromic and Sūdoku Quasigroups
Two quasigroup identities of importance in combinatorics, Schroeder’s Second Law and Stein’s Third Law, share many common features that are incorporated under the guise of palindromic quasigroups. A graph-theoretical technique yields a topological proof for the congruence restrictions on the spectrum of Schroeder or outer palindromic quasigroups. The potential for a comparable proof applicable ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010