$h$ is non-identity element whose square is identity $\implies\langle h\rangle$ is normal

Question

$h$ is non-identity element whose square is identity $\implies\langle h\rangle$ is normal

If the whole group is $G$ and $h\in G$ with the property above then $\langle h\rangle=\{1,h\}$. Clearly $1$ commutes with all elements, hence to check is: $ghg^{-1}=h$

EDIT

OK, You say that the statement is false, but can you translate what the first sentence on page $44$ means (it begins with ''Since the condition...'')

Answer 1

In general the statement is false. What is true is the following. Let $g \in G$, $g \neq 1$, and $g^2=1$. Then $\langle g \rangle$ is normal if and only if $g \in Z(G)$, the center of $G$.