Consider the subset $Q$ of the quaternions defined by $$Q=\{1,-1,i,-i,j,-j,k,-k\}.$$
Show that $Q$ is a group under quaternion multiplication.
I know to prove something's a group, you must show closure, associativity, identity, and inverses. So is the identity is $1$? Is each element is its own inverse? I'm not sure of the operation. I don't remember ever going over quaternions in class. Any guidance will be gratefully appreciated!