# Finding a presentation of the quarternion group. (When do I know if I have given enough relations?)

I was working through presentation of the quaternion group (with element $$8$$), and I let $$a = i$$ and $$b = j$$. I immediately said $$a^4 = b^4 = 1$$, and $$ab^2 a = 1$$.

Since I have a relation for each generator and between the generator, I figured I have the whole presentation. However, when I looked up the presentation of the quaternion group, it was given as

$$Q=\langle F\{a,b\}\mid a^4=b^4=a^2b^2=1 , b^{-1} a d = a^{-1}\rangle.\tag{1}$$

It is hard for me to see whether my initial third relation is a mixture of 3rd or 4th relation given by $$(1)$$.

Also, when do I know if I have given enough relations? Do I have to just write it down and see?

Finding a presentation of a group seems quite tedious!

• You cannot talk about the presentation, because there are lots of presentations, but you do not have enough relations. In fact $b^4=1$ is a consequence of $a^4=1$ and $ab^2a=1$, so one of your relations is redundant. If you adjoined the extra relation $a^{-1}ba=b^{-1}$ then you would have a complete presentation. BTW, your displayed presentation for $Q$ doesn't make sense, because $c$ and $d$ are undefined. – Derek Holt Oct 2 '14 at 7:56
• Sorry I meant $c =a$ &$d=b$ – Quantization Oct 2 '14 at 8:25