Abstract
Let G be a finite group. In this paper, we study how certain arithmetical conditions on the conjugacy class lengths of real elements of G influence the structure of G. In particular, a new type of prime graph is introduced and studied. We obtain a series of theorems which generalize some existed results.
Highlights
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The set of primes dividing the length of some C ∈ con( G ) and there is an edge between two vertices p and q if and only if the product pq divides the length of some C ∈ con( G )
In Theorem 4, we prove that, with only real classes of primary elements being considered, this conjecture is held under some condition on the Sylow 2-subgroups
Summary
It follows from ([9], Theorem B) that, if the prime graph Γr ( G ) of G is disconnected, 2 must be a vertex of Γr ( G ). These facts confirm repeatedly the importance and the special position of the prime 2 in the study of real classes in groups. Based on the above observations, in this article, we consider the set Re∗ ( G ) for G, which are defined earlier, and introduce the prime graph Γr∗ ( G ) related to the classes of.
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