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| Paper IPM / M / 8018 |
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| Abstract: | |||||
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Let n > 0 be an integer and χ be a class of groups. We say
that a group G satisfies the condition (χ,n) whenever in
every subset with n+1 elements of G there exist distinct
elements x,y such that 〈x,y〉 is in χ. 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[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.
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