Thread: conjugate, central element and Normal subgroup

Can you help me to prove this,

Let G be a finite Group, and (a € G). Prove that if a has two conjugates in G, then Z(a) normal subgroup of G.

( Z(a) is the central of a )

Originally Posted by alper82

Can you help me to prove this,

Let G be a finite Group, and (a € G). Prove that if a has two conjugates in G, then Z(a) normal subgroup of G.

( Z(a) is the central of a )

Have you come across the orbit-stabiliser theorem? If so, then use this, and remember that a subgroup of index 2 is normal...