Question

Let $\text{N}$be a normal subgroup of $G$, $a\in G$, and$C$ the conjugacy class of $a$ in$G$ .

Prove that $a\in N$ if and only if$C⫅$$N$ .

Short answer

It is proved that ,$a\in N$ if and only if .

Step-by-step solution

Given that

Given that is a normal subgroup of , and is the conjugacy class of in .

Prove the statement

Let .

Then, .

Therefore, .

Prove only if the statement

Let .

Since is the conjugacy class of in therefore, .

It has been concluded that if and only if .