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Let | b G | = m > n = | a G | be the two longest sizes of the non-central conjugacy classes of G.
Let m > n > 1 be the two longest sizes of the non-central p-regular conjugacy classes of G. Suppose that ( m, n ) = 1 and p is not a prime divisor of n.
The purpose of this paper is to investigate some properties of N under the condition that the two longest sizes of the non-central p-regular G-conjugacy classes of N are coprime.
If m = | b G | > | a G | = n are the two longest sizes of the non-central p-regular G-conjugacy classes of N with ( m, n ) = 1 and n dividing | N / Z ( N ) |, where a, b ∈ N, then either a p-complement of N / Z ( N ) is a prime power order group or (i) | N | p ′ = | a N | p ′ | b N | p ′ | Z ( N ) | p ′ ; (ii) | x G | = m for any non-central p-regular element x ∈ C N ( b ).
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CEO of Professional Science Editing for Scientists @ prosciediting.com