find about center of G s.t H is normal subgroup of order 2

Question

Let G be a finite group and H be normal subgroup of order 2. Then order of center of G is

1. 0
2. 1
3. Even integer $\ge $2
4. Odd integer $\ge $3

I tried this problem by taking G as $S_3$ and H as $ A_3$, therefore as we know center of $ S_n$ for any n is 1. Therefore answer is 1.

But in book answer is option 3. Is there any way to find center and its order under the conditions that I am missing?

Answer 1

A group of order $2$ contains one nonidentity element. For $H$, call this element $x$. Then for all $y\in G$ we have $yxy^{-1}=x$ since $H$ is normal. Thus $yx=xy$, so $x$ commutes with every element of $G$, hence is in the center. Since the center has an element of order $2$, it must be of even order.