# Prove that $Q$ is a group under quaternion multiplication

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!

• You have not had the multiplication of the quaternions defined? – Tobias Kildetoft May 10 '14 at 18:27
• No. This is a study guide for my exam and that's the prompt verbatim. – allie May 10 '14 at 18:33
• What about in the course material? – Tobias Kildetoft May 10 '14 at 18:33
• @allie and you tried to use the internet to look up what quaternions are and didn't find any answer? The wiki tells you how to multiply them, for example. – rschwieb May 10 '14 at 18:37
• Subgroups inherit associativity from the group so you never need to check that for a subgroup. But there will be loads of sources online that explain the quaternions as a group and you'll do this question fine once you have understood them. – EHH May 10 '14 at 19:08