Question

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

Prove that $\mathit{a}\mathbf{\in }\mathit{N}$ if and only if $\mathit{C}\mathbf{\subseteq }\mathit{N}$.

Short answer

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

Step-by-step solution

Given that

Given that N is a normal subgroup of G, $a\in G$and C is the conjugacy class of a in G .

Prove the statement

Let $a\in N$.

Then, $x^{-1}ax\in x^{-1}Nx=N$.

Therefore, $C\subseteq N$.

Prove only if the statement

Let $C\subseteq N$.

Since C is the conjugacy class of a in G , therefore, $a\in N$.

It has been concluded that, $a\in N$if and only if $C\subseteq N$.