Question

* Prove that conjugate relation is an
relation in G.
equivalence​

Answer 1

Answer:

Theorem: Conjugacy is an equivalence relation in a group.

Step-by-step explanation:

Proof:

(i) Reflexivity: Let a∈G, then a=e–1ae, hence a∼a∀a∈G, i.e. the relation of conjugacy is reflexive.

(ii) Symmetric: Let a∼b so that there exists an element x∈G such that a=x–1bx,a,b∈G. Now

a∼b⇒a=x–1bx⇒xa=x(x–1bx)⇒xax–1=(xx–1)b(xx–1)⇒b=xax–1⇒b=(x–1)–1ax–1,x∈G⇒b∼a

Thus a∼b=b∼a. Hence the relation is symmetric.

(iii) Transitivity: Let there exist two elements x,y∈G such that a=x–1bx and b=y–1cy for a,b,c∈G. Hence a∼b, b∼c

⇒a=x–1bxand⇒b=x–1cx⇒a=x–1(y–1cy)x⇒a=(x–1y–1)c(yx)⇒a=(yx)–1c(yx)

Here yx∈G and G are the group. Thereforea∼b,b∼c⇒a∼c.

Hence the relation is transitive.

Thus conjugacy is an equivalence relation on G.