Example of a group in which the equation $x^2=e$ has more than two solutions

Question

I am looking for an example of a group in which the equation $x^2=e$ has more than two solutions, where $e$ is the identity element.

Groups with two solutions are easy to find:

• nonzero reals under multiplication
• cyclic group $\mathbb Z/2\mathbb Z$ under addition
• more generally, cyclic group of even order

But none of these have more than two solutions.

Answer 1

What about the group $\;C_2\times C_2\;,\;\;C_2=$ the cyclic group of order two ?

Answer 2

Permutation group $S_n$ has a lot of solutions for such equation. For example, any transposition will work.

Answer 3

Consider Example:

Suppose $G =D_3$ is the dihedral group of order 6 (gp. of symmetries of an equilateral triangle).Then, as we know, there are exactly three rotations$(R_0,R_{120},R_{240})$ and exactly three reflections ($p_1,p_2,p_3$). Also $R_0=e$ is the identity element and ${R_0}^2=R_0=e,{p_1}^2=e,{p_2}^2=e$ and ${p_3}^2=e$.

Answer 4

In the group of 2x2 invertible matrices, consider the diagonal matrices $\mbox{diag}(1,-1)$, and $\mbox{diag}(-1,-1)$, $\begin{pmatrix} 0 & -1\\ -1 & 0\end{pmatrix}$. Find plenty more by using the inverse of $\begin{pmatrix} a & b\\ c & d\end{pmatrix}$.