Question

If K is a normal subgroup of order 2, in a group G, prove that $K\subseteq Z\left(K\right)$.[Hint: if$K=\left\{e,k\right\}$ and$a\in G$ , what are the possibilities for $\left.ak{a}^{-1}?\right]$

Short answer

It is proved that$K\subseteq Z\left(K\right)$.

Step-by-step solution

 Step 1: Definition of a normal subgroup

A subgroup N of group G is said to be normal if $Na=aN$ for every

$Na=aN$

Proving that  K⊆  ZK

According to the hint, let’s take $K=\left\{e,k\right\}$ and $a\in G$

Since K is a normal subgroup of G, for any element in G,

We can write $ak{a}^{-1}\subset K$.

Since $K=\left\{e,k\right\}$

Therefore, $ak{a}^{-1}=e$or $ak{a}^{-1}=k$

This is possible only when $ak{a}^{-1}=a$because the order of K is 2.

Which Implies $ak=ka$

.Hence, from the definition,$K\subseteq Z\left(K\right)$.