To which group is this presentation isomorphic? $$\left\langle x_1, x_2, x_3 \,\Big\vert\, x_1^2, x_2^2, x_3^2, (x_1 x_3)^2, (x_1 x_2)^2, (x_2 x_3)^2 \right\rangle$$ is a group presentation. Could anyone tell me what does this presentation stand for? 
Source: it is a brief quiz our instructor suggested to us during class, and nobody answered. I can't think of anything myself either.
 A: For example:
$$(x_1x_2)^2=1\iff x_1x_2x_1x_2=1\iff x_1x_2=(x_1x_2)^{-1}=x_2^{-1}x_1^{-1}=x_2x_1$$
So you have here an abelian group generated by three involutions...not many options.
A: The first three relators give that $x_1, x_2, x_3$ are all their own inverses. Then, we may write the latter three relators as
$$x_i x_j x_i^{-1} x_j^{-1}, \qquad (i, j) \in \{(1, 2), (1, 3), (2, 3)\}$$
which says precisely that each pair $(x_i)$ of elements commutes, and so the group is abelian. So, we may write any element as $x_1^a x_2^b x_3^c$, and using the first three relators we may assume $a, b, c \in \{0, 1\}$. However, all of the elements occur in each relator an even numbers of times, so each of the eight elements
$$1, x_1, x_2, x_3, x_1 x_2, x_1 x_3, x_2 x_3, x_1 x_2 x_3$$
is distinct, so the group has order $8$. All nonidentity elements have order $2$, so the group is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$.
A: This is an example of the marvelous theory of Coxeter groups. If you have experience with Coxeter groups then you can realize quite quickly that this is the group of symmetries of the sphere generated by reflections in three great circles intersecting at right angles.
A: consider 5 tiles arranged in a plus sign. Each tile is red on one side and blue on the other. Then let $x_1$ represent flipping over the tiles in the vertical bit, $x_2$ represent flipping over the tiles in the horizontal bit, and $x_3$ represent flipping just the middle tile. Then you can easily show that this setup satisfies the requirements of your presentation. Though @Travis' answer is probably closer to what your instructor was looking for, sometimes it's valuable to actually come up with a concrete version of the group.
