Few associative triples, isotopisms and groups

Few associative triples, isotopisms and groups Let Q be a quasigroup. For $$\alpha ,\beta \in S_Q$$ α , β ∈ S Q let $$Q_{\alpha ,\beta }$$ Q α , β be the principal isotope $$x*y = \alpha (x)\beta (y)$$ x ∗ y = α ( x ) β ( y ) . Put $$\mathbf a(Q)= |\{(x,y,z)\in Q^3;$$ a ( Q ) = | { ( x , y , z ) ∈ Q 3 ; $$x(yz)) = (xy)z\}|$$ x ( y z ) ) = ( x y ) z } | and assume that $$|Q|=n$$ | Q | = n . Then $$\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})$$ ∑ α , β a ( Q α , β ) / ( n ! ) 2 = n 2 ( 1 + ( n - 1 ) - 1 ) , and for every $$\alpha \in S_Q$$ α ∈ S Q there is $$\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2$$ ∑ β a ( Q α , β ) / n ! = n ( n - 1 ) - 1 ∑ x ( f x 2 - 2 f x + n ) ≥ n 2 , where $$f_x=|\{y\in Q;$$ f x = | { y ∈ Q ; $$y = \alpha (y)x\}|$$ y = α ( y ) x } | . If G is a group and $$\alpha$$ α is an orthomorphism, then $$\mathbf a(G_{\alpha ,\beta })=n^2$$ a ( G α , β ) = n 2 for every $$\beta \in S_Q$$ β ∈ S Q . A detailed case study of $$\mathbf a(G_{\alpha ,\beta })$$ a ( G α , β ) is made for the situation when $$G = \mathbb Z_{2d}$$ G = Z 2 d , and both $$\alpha$$ α and $$\beta$$ β are “natural” near-orthomorphisms. Asymptotically, $$\mathbf a(G_{\alpha ,\beta })>3n$$ a ( G α , β ) > 3 n if G is an abelian group of order n. Computational results: $$\mathbf a(7) = 17$$ a ( 7 ) = 17 and $$\mathbf a(8) \le 21$$ a ( 8 ) ≤ 21 , where $$\mathbf a(n) = \min \{\mathbf a(Q);$$ a ( n ) = min { a ( Q ) ; $$|Q|=n\}$$ | Q | = n } . There are also determined minimum values for $$\mathbf a(G_{\alpha ,\beta })$$ a ( G α , β ) , G a group of order $$\le 8$$ ≤ 8 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Designs, Codes and Cryptography Springer Journals

Few associative triples, isotopisms and groups

, Volume 86 (3) – Feb 11, 2017
14 pages

/lp/springer_journal/few-associative-triples-isotopisms-and-groups-0Y6tqrWK27
Publisher
Springer Journals
Subject
Mathematics; Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits
ISSN
0925-1022
eISSN
1573-7586
D.O.I.
10.1007/s10623-017-0341-9
Publisher site
See Article on Publisher Site

Abstract

Let Q be a quasigroup. For $$\alpha ,\beta \in S_Q$$ α , β ∈ S Q let $$Q_{\alpha ,\beta }$$ Q α , β be the principal isotope $$x*y = \alpha (x)\beta (y)$$ x ∗ y = α ( x ) β ( y ) . Put $$\mathbf a(Q)= |\{(x,y,z)\in Q^3;$$ a ( Q ) = | { ( x , y , z ) ∈ Q 3 ; $$x(yz)) = (xy)z\}|$$ x ( y z ) ) = ( x y ) z } | and assume that $$|Q|=n$$ | Q | = n . Then $$\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})$$ ∑ α , β a ( Q α , β ) / ( n ! ) 2 = n 2 ( 1 + ( n - 1 ) - 1 ) , and for every $$\alpha \in S_Q$$ α ∈ S Q there is $$\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2$$ ∑ β a ( Q α , β ) / n ! = n ( n - 1 ) - 1 ∑ x ( f x 2 - 2 f x + n ) ≥ n 2 , where $$f_x=|\{y\in Q;$$ f x = | { y ∈ Q ; $$y = \alpha (y)x\}|$$ y = α ( y ) x } | . If G is a group and $$\alpha$$ α is an orthomorphism, then $$\mathbf a(G_{\alpha ,\beta })=n^2$$ a ( G α , β ) = n 2 for every $$\beta \in S_Q$$ β ∈ S Q . A detailed case study of $$\mathbf a(G_{\alpha ,\beta })$$ a ( G α , β ) is made for the situation when $$G = \mathbb Z_{2d}$$ G = Z 2 d , and both $$\alpha$$ α and $$\beta$$ β are “natural” near-orthomorphisms. Asymptotically, $$\mathbf a(G_{\alpha ,\beta })>3n$$ a ( G α , β ) > 3 n if G is an abelian group of order n. Computational results: $$\mathbf a(7) = 17$$ a ( 7 ) = 17 and $$\mathbf a(8) \le 21$$ a ( 8 ) ≤ 21 , where $$\mathbf a(n) = \min \{\mathbf a(Q);$$ a ( n ) = min { a ( Q ) ; $$|Q|=n\}$$ | Q | = n } . There are also determined minimum values for $$\mathbf a(G_{\alpha ,\beta })$$ a ( G α , β ) , G a group of order $$\le 8$$ ≤ 8 .

Journal

Designs, Codes and CryptographySpringer Journals

Published: Feb 11, 2017

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