Certain numerical results in non-associative structures

Some preliminaries on groups are needed. Let G be a finite group and n≥2 be a positive integer. We consider the n th-commutativity set of G as the following:

Proposition 2.1

Let G be a finite non-abelian group and n≥2 be a positive integer. If M=M(G,2) is a finite Moufang loop, then

Moreover, if G be of odd order. Then

Two special cases of G where G=D2m , the dihedral group of order 2 m and G=Q2m , the generalized quaternion group of order 2m , give us two infinite classes of Moufang loops M(D2m,2) and M(Q2m,2) , where m≥3 is a positive integer. In fact, we prove the following results about these classes of loops:

Proposition 2.2

Let M=M(D2m,2) be a finite Moufang loop, where m≥3 , and n be an odd integer and d=g.c.d(m,2n) . Then,

Proposition 2.3

Let M=M(Q2m,2) be a finite Moufang loop, where m≥3 , and n be an integer and d=g.c.d(2m-1,2n) . Then,

At first, we give certain preliminary results as in the following lemmas:

Lemma 2.4

Let D2m be a dihedral group, where m≥3 , and n be an odd integer, d=g.c.d(m,n) and r=md . Then

Proof

Let m be an odd integer and we know that D2m=A∪B , where A={1,a,⋯,am-1} and B={b,ab,⋯,am-1b} . Clearly |An|=|A|d=md=r , and the element 1∈An commutes with all the elements of D2m and each other element of An commutes with all the elements 1,a,⋯,am-1∈D2m . This implies that there are d×(2m+(r-1)m)=m2+m2r pairs as (a,g)∈A×D2m such that ang=gan . Also Bn={(ajb)n∣ajb∈B,0≤j≤m-1}=B and each element of B commutes with two elements 1 and itself. So there are 2 m pairs as (b,g)∈B×D2m such that bng=gbn . Consequently,

Lemma 2.5

Let D2m be a dihedral group, where m≥3 , and n be an even integer, d=g.c.d(m,n) and r=md . Then,

Proof

Let r be an even integer and we know that D2m=A∪B . Clearly |An|=|A|d=md=r and the elements 1,ar∈An commute with all the elements of D2m and each other element of An commutes with the elements 1,a,⋯,am-1∈D2m . This implies that there are d×(2(2m)+(r-2)m)=mmr(r+2) pairs as (a,g)∈A×D2m such that ang=gan . Also Bn={(ajb)n∣ajb∈B,0≤j≤m-1}=B and each element of B commutes with all elements of D2m . So there are 2m|B|=2m2 pairs as (b,g)∈B×D2m such that bng=gbn . Consequently,

Lemma 2.6

Let Q2m be a generalized quaternion group, where m≥3 , and n be an odd integer. Then,

Proof

Let n be an odd integer and note that Q2m=A∪B , where A={1,a,⋯,a2m-1-1} and B={b,ab,⋯,a2m-1-1b} . Clearly An={ajn∣aj∈A,0≤j≤2m-1-1}=A . Obviously the elements 1,a2m-2∈An are central elements in Q2m and each other element of An commutes with all the elements 1,a,⋯,a2m-1∈Q2m . So there are 2(2m)+(2m-1-2)2m-1=2m(1+2m-2) pairs as (a,g)∈A×Q2m such that ang=gan . Also Bn=B and each element of B commutes with the four elements in Q2m . This shows that there are 4(2m-1) pairs as (a,g)∈B×Q2m such that bng=gbn . Hence,

Lemma 2.7

Let Q2m be a generalized quaternion group, where m≥3 , and n be an even integer. Then,

Proof

If n≡0(mod2m-2) , then Q2mn={gn∣g∈Q2m}={1} . So,

By results of these lemmas, we get explicit values for Pn(D2m) and Pn(Q2m) as in the following:

Lemma 2.8

Let D2m be a dihedral group, where m≥3 , and n be an odd integer, d=g.c.d(m,n) and r=md . Then,

Proof

By Lemma ^2.4 , and by definition of Pn(D2m)=|Cn(D2m)|(2m)2 , we deduce that

Lemma 2.9

Let D2m be a dihedral group, where m≥3 , and n be an even integer, d=g.c.d(m,n) and r=md . Then,

Proof

By Lemma ^2.5 , and by definition of Pn(D2m)=|Cn(D2m)|(2m)2 , we get the result

Lemma 2.10

Let Q2m be a generalized quaternion group, where m≥3 , and n be an odd integer. Then,

Proof

By Lemma ^2.6 , and by definition of Pn(Q2m)=|Cn(Q2m)|(2m)2 , we deduce that

Lemma 2.11

Let Q2m be a generalized quaternion group, where m≥3 , and n be an even integer. Then,

Proof

By Lemma ^2.7 , and the definition of Pn(Q2m)=|Cn(Q2m)|22m , we get the result

To prove the main propositions, we recall the following properties of the Moufang loop M=M(G,2) , where G is a given group. These may be verified easily by considering the binary operation in M : (i)

(gu)ohn=hno(gu) if and only if h2n=1;

Proof of Proposition 2.1

Let T={(g,h)∣g,h∈M,ghn=hng} . Simplify the binary operation o in M as the table

Since Pn(M)=|T|(2|G|)2 , then it is sufficient to enumerate | T |. For every pair (g,h)∈M , there are four cases: Case 1 :

g,h∈G . In this case, there are |Cn(G)| number of elements of the type ( g ,  h ) in T .

Consequently, T has contains |Cn(G)|+2α|G|+β|G| number of elements all together. Hence,

Proof of Proposition 2.2

By Lemma ^2.8 , and Proposition ^2.1 , we get

Proof of Proposition 2.3

By Proposition ^2.1 ,

The following remarks are the results of Proposition ^2.3 . They give us useful upper bounds for Pn(M) where M=M(Q2m,2) .

Remark 3.1

For all integers, n≥2 and m≥3 let M=M(Q2m,2) and d=g.c.d(2m-1,2n) . Then, ( i )

For odd values of n , Pn(M)≤132(9+d) ,

Remark 3.2

For distinct odd primes p1,⋯,pk and positive integers α1,⋯,αk , if n=p1α1p2α2⋯pkαk , then

Proof

For n=p1α1p2α2⋯pkαk , we use Remark ^3.1 -(i) to get d=g.c.d(2m-1,2n)=2 . So,