(Dummit's AA, 1.5, P3) Are these presentations of the Quarternion group equivalent? For example, from wiki, we know that 
$$ \langle i, j \mid i^4 =1, i^2 = j^2, j^{-1}ij = i^{-1} \rangle = Q $$
where $Q$ denotes Quaternion group.
And by my own inspection, I speculated that
$$ \langle i,j \mid i^4 = j^4 = 1, ij = j^3i \rangle =Q $$
though I'm not really sure if it is correct.
I've checked every relation that hold in $Q$ can be derived from the relations in the presentations. But finding the order of the groups generated by the above presentations, I couldn't do it rigorously nor systematically. (I've considered every element that can be formed by the generators in the form $i^a j^b$ with $0 \leq a, b \leq 3$ and relations in the second presentation and for everything I've checked which is equal to which. And for all the elements in that form, to prove which is not equal to which, I've supposed which is equal to which and derived a contradiction. But personally I think my method is just a mess.)
Is there any effective, systematic way for proving this without checking everything? 
And please correct me if there is anything wrong.
 A: Here is a spooky short presentations of $Q_8$
$$\langle i,j : iji=j, i^2=j^2 \rangle$$
For a finite group, the number of relations is always at least equal to the number of generators, so 2 is the minimum number of relations with two generators. In fact, a group such as $D_8$ cannot have a presentation with only two relations (hard to prove at this stage), because there is a group $G$ ($D_{16}$ in fact) with $G/(Z(G) \cap [G,G]) \cong D_8$. This obstruction to having a short presentation is known as the Schur multiplier.
As far as systematic methods, basically no in general. Standard practical methods for problems of this size are coset enumeration and the Knuth-Bendix algorithm. Sims has a nice book, Computations with Finitely Presented Groups that I found quite enlightening.
If you start with a presentation for a group, you can mix it up using Tietze transformations which can add/remove superfluous generators, and length/shorten relations using other relations. It is fairly similar to Gaussian elimination, except you only get the row ops, you rarely actually know when you have reached the canonical form. In particular, we know there is no algorithm to use Tietze transformations to prove a trivial group is trivial, but we do know at least there is SOME way to use them to prove a trivial group is trivial.
A: This one is based on using the graphs in presentation of a group, the method called Van Kampen diagram:

You see we can find the presentation of second group as $j^4=1$ the outer circular path and $j^3ij^{-1}i^{-1}=1$ as the closed path $ABCDEFA$.
