I’m reading the book “Geometric group theory: an introduction”, written by Clara Loh. In the introduction part there’s the following exercise about presentations of groups (exercise 2.E.24):
Prove that $$\langle s_1,s_2,s_3 \mid (s_1s_2)^2,(s_1s_3)^2, (s_2s_3)^2\rangle\cong (\Bbb Z/2)^3$$
I tried to prove that the generators need to have order two, but I wasn’t able to. I thought that maybe it could be useful two prove that the $s_is_j$’s commute with the each other and then show that these products generate the entire group.