2005/11/28 by Aliréza Abdollahi, Alireza Abdollahi, Abdollahi, Alireza +3
Computer Science · Engineering · Mathematics · #20F45 #20F99 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems #math.GR #msc:20F45 #msc:20F99
paper · pdf · doi:10.48550/arxiv.math/0511667
Published in Bulletin of the Iranian Mathematical Society, Vol. 30 No. 2 (2004), pp. 1-20
arxiv created 2005/11/28 · openalex publication_date 2005/11/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let n>0 be an integer and X be a class of groups. We say that a group G satisfies the condition (X,n) whenever in every subset with n+1 elements of G there exist distinct elements x,y such that <x,y> is in X. Let N and A be the classes of nilpotent groups and abelian groups, respectively. Here we prove that: (1) If G is a finite semi-simple group satisfying the condition (N,n), then |G|<c^2[log21n]n2 [log21n]!, for some constant c. (2) A finite insoluble group G satisfies the condition (N,21) if and only if (G)/(Z^*(G))≅ A5, the alternating group of degree 5, where Z^*(G) is the hypercentre of G. (3) A finite non-nilpotent group G satisfies the condition (N, 4) if and only if (G)/(Z^*(G))≅ S3, the symmetric group of degree 3. (4) An insoluble group G satisfies the condition (A,21) if and only if G≅ Z(G)× A5, where Z(G) is the centre of G. (5) If d is the derived length of a soluble group satisfying the condition (A,n), then d=1 if n∈ \1,2\ and d≤ 2n-3 if n≥ 2.