On an extension of a theorem on conjugacy class sizes |
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Authors: | Qingjun Kong Xiuyun Guo |
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Institution: | 1.Department of Mathematics,Tianjin Polytechnic University,Tianjin,People’s Republic of China;2.Department of Mathematics,Shanghai University,Shanghai,People’s Republic of China |
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Abstract: | Let G be a finite group. We extend Alan Camina’s theorem on conjugacy classes sizes which asserts that if the conjugacy classes
sizes of G are {1, p
a
, q
b
, p
a
q
b
}, where p and q are two distinct primes and a and b are integers, then G is nilpotent. We show that let G be a group and assume that the conjugacy classes sizes of elements of primary and biprimary orders of G are exactly {1, p
a
, n,p
a
n} with (p, n) = 1, where p is a prime and a and n are positive integers. If there is a p-element in G whose index is precisely p
a
, then G is nilpotent and n = q
b
for some prime q ≠ p. |
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Keywords: | |
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