Subgroups Which Are the Union of Three Conjugate Classes

The power graph on the conjugacy classes of a finite group

“…It follows that G ′ = Z(G) and G/Z(G) is an elementary abelian 2-group. Now, suppose that Z(G) = 1, then by Theorem 2.1 of [15], G = G ′ N is a Frobenius group with kernel G ′ that is an elementary abelian p-group and its complement N that is abelian such that |G| = |G ′ |(|G ′ | − 1). Set |G ′ | = p r and so |G| = p r (p r − 1) = 2 s p r .…”

Subgroups Which Are the Union of Four Conjugacy Classes

“…Step 3: Suppose the order of every nonidentity element of H is a p-power for some fixed prime p. Then H is a p-group of order p [2] , H [2] ≤ H [4] = 1.…”

On the normal subgroup with exactly two G-conjugacy class sizes

“…Since then, many authors have investigated the relationship between the structure of a group and its conjugacy class sizes (for example, [1,2,4,5,[8][9][10][11][12][13][14]). Among these results, a classic result by Itô [8] asserts that a group G is nilpotent if |x G | = 1 or m for every x ∈ G. Recently, Beltrán and Felipe [2] proved that every Hall p -subgroup of a p-solvable group is nilpotent if |x G | = 1 or m for every p -element x of G. On the other hand, the structure of a normal subgroup N of a group G was given if N is the union of some G-conjugacy classes (see [9][10][11][12]). Now, we are interested in the following question: Let G be a finite group and let N be a normal subgroup of G. If |x G | = 1 or m for every element x ∈ N , is N nilpotent?…”