Certain numerical results in nonassociative structures
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Abstract
The finite noncommutative and nonassociative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer \(n\ge 2\), the nthcommutativity degree of a finite algebraic structure S, denoted by \(P_n(S)\), is the probability that for chosen randomly two elements x and y of S, the relator \(x^ny=yx^n\) holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the nthcommutativity degree of two infinite classes of finite loops, which are noncommutative and nonassociative. Also by deriving explicit expressions for nthcommutativity degree of these loops, we will obtain best upper bounds for this probability.
Keywords
Loop Moufang loop nthcommutativity degreeMathematics Subject Classification
11B39 20P05 20N05Introduction
Following [9], recall the definition of nthcommutativity degree of an algebraic structure S, denoted by \(P_{n}(S)\), which is the probability that for two elements x and y of S the relator \(x^ny=yx^n\) holds, for a given integer \(n\ge 2\). This degree has been studied mainly for associative algebraic structures during the years, one may consult Lescot [2], for example. We here intend to consider two classes of nonassociative algebraic structures and calculate their nthcommutativity degree, for every integer \(n\ge 2\). Our considered classes of loops are \(M(D_{2m},2)\) and \(M(Q_{2^m},2)\) where, for every \(m\ge 3\), \(D_{2m}\) is the dihedral group of order 2m and \(Q_{2^m}\) is the generalized quaternion group of order \(2^m\). Since these groups are nonabelian, the considered loops are nonassociative loops. The nthcommutativity degree of these loops will be calculated in “Main results” section, and the “Conclusion” section is devoted to study of upper bounds for the calculated probabilities.
To clear the terminology “nthcommutativity” we have to mention the difference between this definition and that of Lescot [2] where, he used this terminology in the study of multiple commutativity degree of groups.
Main results
Proposition 2.1
Two special cases of G where \(G=D_{2m}\), the dihedral group of order 2m and \(G=Q_{2^{m}}\), the generalized quaternion group of order \(2^{m}\), give us two infinite classes of Moufang loops \(M(D_{2m},2)\) and \(M(Q_{2^{m}},2)\), where \(m\ge 3\) is a positive integer. In fact, we prove the following results about these classes of loops:
Proposition 2.2
Proposition 2.3
At first, we give certain preliminary results as in the following lemmas:
Lemma 2.4
Proof
Lemma 2.5
Proof
Lemma 2.6
Proof
Lemma 2.7
Proof
By results of these lemmas, we get explicit values for \(P_{n}(D_{2m})\) and \(P_{n}(Q_{2^{m}})\) as in the following:
Lemma 2.8
Proof
Lemma 2.9
Proof
Lemma 2.10
Proof
Lemma 2.11
Proof
 (i)

\((gu)oh^{n}=h^{n}o(gu)\) if and only if \(h^{2n}=1;\)
 (ii)

\((gu)^{n}oh=ho(gu)^{n}\) if and only if \(h^{2}=1;\)
 (iii)

\((gu)^{n}o(hu)=(hu)o(gu)^{n}\) if and only if \((g^{1}h)^{2}=1;\)
 (iv)

if \(g^{n}h=hg^{n}\) then \((gu)^{n}o(hu)=(hu)o(gu)^{n}\) if and only if \(h^{2}=g^{2}\).
Proof of Proposition 2.1
o  G  Gu 
G  G *G  G * Gu 
Gu  Gu * G  Gu * Gu 
 Case 1:

\(g, h\in G\). In this case, there are \(C_{n}(G)\) number of elements of the type (g, h) in T.
 Case 2:

\(g\in Gu\) and \(h\in G\). Then, \(g=g_{1}u\), \(g_{1}\in G\). By (ii), we conclude that \(h^{2}=1\). So, there are exactly \(\alpha Gu=\alpha G\) number of elements in T of the type \((g_{1}u, h)\).
 Case 3:

\(g\in G\) and \(h\in Gu\). So, \(h=h_{1}u\) where \(h_{1}\in G\). In this case, we use (i) and deduce that there are \(\beta G\) number of distinct elements in T of the type \((g, h_{1}u)\), because of the validity of the relation \(h_{1}^{2n}=1\).
 Case 4:

\(g\in Gu\) and \(h\in Gu\). So, \(g=g_{1}u\) and \(h=h_{1}u\), for \(g_{1}, h_{1}\in G\). By using (iii), we confirm the existence of \(\alpha G\) number of distinct elements in T of the type \((g_1u,h_1u)\), for, \((g_{1}^{1}h_{1})^{2}=1\).
Proof of Proposition 2.2
Proof of Proposition 2.3
Conclusion
The following remarks are the results of Proposition 2.3. They give us useful upper bounds for \(P_{n}(M)\) where \(M=M(Q_{2^{m}},2)\).
Remark 3.1
 (i)

For odd values of n, \(P_{n}(M)\le \frac{1}{32}(9+d)\),
 (ii)
 For even values of n,$$\begin{aligned} \left\{ \begin{array}{lll} P_{n}(M)\le \frac{1}{32}\left( 16+d\right) , &{} &{} n\equiv 0\,(mod\, 2^{m2}),\\ &{} &{}\\ P_{n}(M)\le \frac{1}{32}\left( 15+d\right) , &{} &{} n\equiv \pm 2^{r}\,(mod\, 2^{m2}), ~r=1,2,\dots ,m3. \end{array} \right. \end{aligned}$$
Remark 3.2
Proof
Notes
Acknowledgements
The authors would like to thank the referees for a very careful reading of the paper and complete comments and useful suggestions, which improved considerably the presentation of this paper.
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