Finite groups with nilpotent centralizers

Finite group with Hall coverings

“…The soluble CN -groups are known (see [7,Theorem 14.1.5]), while Suzuki proved that a simple C/V-group is isomorphic to one of the following list (see [12, In the same paper [24,Theorem 4], Suzuki proved that a non-soluble CA'-group is a C/r-group, that is a group of even order in which thecentralizerof any involution is [15] Finite group with Hall coverings 15 then |7| = 4 and H = Nc(T) is isomorphic to 5 4 . We can apply lemma 5.2 to the preimages H and 7 of H and 7 in G and /?…”

On finite groups containing three 𝐶𝐶-subgroups

“…Hence, centralizers of elements of order r are Gr groups. Therefore, Gr is a CC-subgroup of G [9]. Since all elements of G* are conjugate, Gr is elementary abelian, and CG(Gr) = Gr.…”

A Characterization of the Zassenhaus Groups

“…Using a equality Σβ(G) = -1@IJ we can easily prove that there exists no involution which is not conjugate to 7. Proof If © (7) is an elementary abelian 2-group, then by Lemma 11, © is a (CIT)-group (Suzuki [11]). If @ has a non trivial solvable normal subgroup, then @ has a regular normal subgroup dl.…”