A concrete example of involution in group theory I am reading the textbook "Introduction to Modern Algebra, Joyce 2017" and in the Cyclic groups and subgroups section, there is a following sentence about involution.

An involution $a$ is an element of a group which is its own inverse, $a^{-1} = a$. Clearly, the order of an involution $a$ is $2$ unless $a = 1$, in which case the order of $a$ is $1$. 

I was trying to come up with a concrete example, but I am having difficulties. Maybe when $a = \frac{1}{2}, \frac{1}{2}^2 = 1$? I also searched the wikipedia page for a concrete example, but wasn't able to find anything.
Can someone provide an example with numbers for me please? I want to understand why the author writes "clearly" that the order of an involution $a$ is $2$. Thank you
 A: Example: in the cyclic group $\mathbb{Z}_{12}$, $6$ is an involution since $6+6=0$ in $\mathbb{Z}_{12}$. More generally, in $\mathbb{Z}_{2n}$, the element $n$ is an involution.
Notice that the requirement $a=a^{-1}$ is equivalent to $aa=1$ (just multiply both sides by $a$). Therefore an involution must satisfy $a^2=1$. So the only way it does not have order $2$ is if it has order smaller than $2$, i.e., $a=1$. 
A: An involution in a group is any non-trivial element of order two (not a subset of them, as the phrase you mention suggests, and also the identity is explicitly omitted). That is, $a$ is an involution if and only if $a\neq1$ and $a^2=1$:
$$
\begin{align*}
a&=a^{-1}\\
\Leftrightarrow a\cdot a&=a\cdot a^{-1}&\text{(left multiply by $a$)}\\
\Leftrightarrow a^2&=1
\end{align*}
$$
For a concrete example of an involution, consider the set $\{0, 1\}$ under addition modulo $2$. Here, the identity is $0$, and $1+1=2=0\pmod2$.
A: An involution is, in essence, anything that undoes itself. For example, flipping a coin over by one axis, the action of turning a key for certain padlocks, or adding one modulo two.
A: Take the general linear group $G=GL_n(K)$ and the element $A=-I_n$. It has order $2$ because of $A^2=I_n$ and hence is an involution.
