Finite Groups with a Certain Number of Elements Pairwise Generating a Non-nilpotent Subgroup
نویسندگان
چکیده
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| < c2[log21 n]n 2 [log21 n]!, 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.
منابع مشابه
Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup
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