Find a group $G$ that has elements $V$ and $H$ both with order $2$ and the order of $VH$ is $3$.

Question

Find a group $G$ that has elements $V$ and $H$ both with order $2$ and the order of $VH$ is $3$.

This question appeared on my final and even though it looks like an easy question I had difficulty finding a group.

Answer 1

$S_3$ gives such an example: Choose $V = (12)$ and $H = (23)$; then $VH$ is a $3$-cycle, and so has order $3$.

---

As a guide for finding this sort of example, it's immediate that the group cannot be abelian: This would mean that $VH$ has order $2$. The group order must be divisible by both $2$ and $3$, so $S_3$ is the smallest possible example. The next smallest non-abelian group satisfying these order conditions has order $12$.

Answer 2

Consider the symmetries of an equilateral triangle ($D_3$ or $D_6$, depending on choice of notation). Reflecting across an axis is an operation of order two, but if we reflect over one axis and then a different axis, we get a rotation of the triangle.