Tu's An Introduction to Manifolds, in question 15.9 asks a question about the center of $GL(2,\mathbb{R})$. He claims it is isomorphic to $\mathbb{R}^*$.
He then states that the group, $(\mathbb{R}^*, \times)$ has two elements of order $2$, which makes no sense to me as $-1$ is the only one I can find.
Can someone confirm I am correct so I can send an update for the errata?
Best Answer
You are correct, $\Bbb R^\times$ under multiplication has a single element of order $2$. In fact, if the number of order-$2$ elements in an abelian group is finite, then that number must be $2^n-1$ for some natural number $n$. So it can't be $2$.
Using my powers of mind reading, I induce that the author might have meant that there are two elements that are their own inverse, which is to say, two elements whose square is $1$.